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Capital budgeting and plant capacity
Engineering Costs and Production Economics, 21 ( 1991 ) 233-241
233
Elsevier
Capital budgeting and plant capacity ltzhak Krinsky Finance and Business Economics, Faculty of Business~ McMaster University, Hamilton. Ontario. Canada L8S 4M4 (Received 18 June, 1990; accepted in revised form 20 November 1990)
Abstract In this paper a model which explicitly recognizes the relationships between the plant capacity, the lead time for building the plant, and the discount rate used in evaluating the project, is developed. It is shown that the introduction of a discount rate which is a function of capacity, substantially alters the results obtained in a previous study where the discount rate is assumed to be constant. In particular, since the net benefits of an additionel unil of capacity is dicounted uz,ing a higher rate, the NPV is suppressed. An extensive sensitivity analyses on the nonmonetaq,' and monetary parameters and their effect on the optimal plant size vs. optima! NPV is provided.
|. Introduction As a result of the high degree of uncertainty, many projects in recent years are characterized by capital expenditure overruns and poor performance. Research reports often verify the general impression that estimates of their performance, tim,~tables, and costs are extraordinarily accident-prone, particularly when new techno!ogies are involved (see, for example, Merrow, Phillips, and Myers [1 ] ). If a company is contemplating a major investment project that involves a new technology, market, or a new size ~o the company, serious overruns are a real possibility. Frequentb ~~ost overruns are a result of an optimistic forecast of the plant's construction time and an unrealistic assessment of the difficulties one might face when exploring new territories. Ross [2 ], in reviewing the capital budgeting process of twelve large manufacturers, points out that even in cases where sensitivity analysis for potential risk was not avoided, the projects' "benefits and economic lives tended to be less than predicted because of unanticipated changes in production rates and technology" (p. 18 ). He adds that these insights were, usually, not incorporated in a formal fashion in the financial analysis.
0167-188X/91/$03.50 © 199 I--Elsevier Science Publishers B.V.
An obvious example is General Motors (GM). In its quest to secure a place in the subcompact car market, GM decided at the beginning of 1985 to spend 3.5 billion dollars to build its Saturn plant, about six times the 600 million dollars it spent to construct its Hamtramch assembly plant in Detroit which is considered one of the world's most technologically advanced. At an original production capacity of half a million cars a year, it was expected that the Saturn plant could build a car in 40 hours of labour (compared to a total of 100 labour hours that Japanese car makers now put). The plant was to be built with an automatic guided vehicle (AGV) system. GM also entered into a joint venture with the Japanese robot maker, Fanuc, to develop various machines to save labour and boost product quality. GM chairman has stated that "Saturn is the key to GM's long term competitiveness, survival, and success as a domestic producer." [ Fisher [ 3 ] p. 8]. The Saturn car project of GM has come a long way since 1985. In a shareholders meeting held in May 1987 GM announced that, due to changes in market conditions, the Saturn plant will not be able to compete with South Korean and other foreign manufacturers. It has also slashed production plans to 250,000 cars a year from
234 500,000, and the labour force to 3,000 from the 6,000 originally announced. In fact, the plant is not scheduled to start production before the end of 1990 and one might question the profitability of the entire project. In this paper, the problem faced by a decision maker described in the previous paragraphs is analysed. Specifically, it deals with the optimal plant capacity in a changing environment where the product's life cycle ends after T periods. Further, the relationships between the plant capacity, the lead time for building the plant, and the discount rate used in evaluating the project, are explicitly recognized. A realistic framework in which the impact of stochastic projects lives on the expected net present value and plant capacity, is also provided. While the model deals with capacity, it is also applicable to cases where the lead time is a function of complexity rather than capacity (i.e., cases where advanced manufacturing technologies are introduced). In what follows, a simple NPV maximization model for a company assumed to be a price taker is developed in Section II. The problem is to choose an optimal plant size so as to maximize the NPV. As sucb, our model is in the spirit of Fuller and Gerchak [ 4 ] and, especially, Krinsky and Parlar [5 ]. The major differences between the results provided here and those in previous studies is the fact that the discoun'., rate is assumed to be a function of capacity. Assuming T as a known parameter and the discount rate p dependent of capacity, we analyze the concavity properties of the NPV. For a wide range of parameters we have tested, NPV was concave. The results are illustrated with several numerical examples. The sensitivity of the optimal solution to various monetary and nonmonetary parameters is also investigated. In Section 3, the end of life cycle T is assumed to be a nonnegative random variable with a known probability density function. The incorporation of a stochastic - - not new. The work by Wagle [6 ], and especially Bey [7,8 ], among others, addresses some aspects of the stochastic project life problem. None, however, deals with the problem within a model which explicitly reco ognizes the relationship between the plant capacity and the lead ;ime to build the plant. The very
complicated nature of the resulting expected NPV function makes it difficult to analyze its qualitative properties. Nevertheless, numerical experiments indicate that this function as the one with deterministic T is concave, at least for the practical range of parameter assumed. The paper is concluded with a summary of findings and a discussion on the implications of the results. 2. The basic model
The basic model is similar to the one introduced by Krinsky and Parlar [ 5 ] in which p is assumed to be a parameter independent of capacity. Let x be the plant size, measured in production capacity per unit of time. Here, x is the decision variable to be chosen at time t = 0. Define the following. L (x) = Construction lead time of plant of size x. Here L ( 0 ) =0, L' (x) >0, L" (x)>~0 and L (x) ~I0 with r(0 ) >I0. It is assumed that r(t) is exogenous to the model and is equal to the estimated mean of some future random process [ r(g) >I0 ]. Note that r(t) becomes relevant only after construction is completed and the plant starts operating. c ( t ) = After tax real construction cost rate per unit of capacity at t >I0 with c(0) >i 0. Also assumed to be exogenous [c(t)>~Ol. So(t) = After tax salvage value per unit of capacity when the plant is completed at t~, I ~<~t <~T. ,o(x) = The appropriate risk-adjusted discount rate where p(x)>O, p ' ( x ) > O and p"(x)>~0. We adopt here the principal of allocating funds using a risk-adjasted hurdle rate (Brealey and Myers [9 ] ). An alternative might be to make r stochastic and a function of the economy. The p could be made endogenous to the model by making p = f ( c o v ( r , economy) ). This alternative, however, produces a complicated model which
235 does not serve the purpose of the paper. Krinsky and Parlar [5] also achieve risk adjustment in a similar model by utilizing a riskadjusted discount rate. The risk level in their model, however, is not a function of plant capacity - it is constant over all capacity levels x. The risk-adjusted discount rate in this model, is assumed to be an increasing function of capacity. This is so because all companies with non-zero beta would find each marginal unit to be sold to be less likely, except in better state outcomes. In other words, the systematic risk of the marginal unit increases with capacity. When studying various energy-saving projects, Ross [2] states that contingencies such as reduced production because of weak sales or changes in the production process are often ignored when the projects are evaluated. He also observes that smaller projects usually had higher hurdle rates despite the fact that there was no reason "that would justify subjecting them to higher hurdle rates" (p. 20). As a result, smaller projects were discriminated against. As shown below, making the discount rate a function of capacity has a major impact on the optimal capacity level. Given the above definitions, the present value (PV) at time t = 0 of the after tax net cash flows is equal to
S(x) = e-~"" ~/-~-"~So( T ) x e
-pc~~ t T - Lcx) 1
=so(T)xe -p~-'~r
(3)
is the PV of the salvage value when the plant is salvaged at time T. Combining ( l ) - ( 3 ) gives the NPV to be equal to NPV(x)
=R(x) -C(x) + S(x)
(4)
which is to be maximized with respect to capacity x. Note that NPV (0) =0, i.e. if the capacity is at zero level, then the NPV is also zero. Next, at the other end point x-x~ =L-~ (T) one can obtain R(xt)=0 "d"
C(x~ ) = Se -''~ ' ~"xl c( u )du o
and
S(xl)
=So( T)xl e -pI'~T
so that 1"
NPV(xl) = -
[
fe-m")'c(u)du
a¢
o
--So( T)e-PC~)r]Xl < 0 since
7"
T
R(x) =e-p'''~Lc'' ~ e-PC"~t'-L{x~]xr(u)du
f
L(x)
e-"~-""c ( u )du> So( T)e -ptx~r
o
T
= ~ e-PC"'"xr(u)du
(l)
L(x)
Also, L(.v)
C ( x ) = e -p~')L~'~
f et'~'~tL~"}-UJxc(u)du o
L(x)
= ~ e-P''}"xc(u)du
(2)
o
is the PV of construction costs incurred between 0 and L (x). Further,
i.e the PV of construction cost of one unit capacity for T time units is greater than the PV of salvage value of that unit capaci.ty due to the fact that no revem~es will be generated. Therefore, NPV (0) = 0 and NPV (xt) < 0 at the two feasible end points of the decision variable. We were unable to prove the concavity of NPV(x) for general p(x) due to the extreme complexity of the second derivative NPV" (x). Nevertheless, in the numerical examples we have conducted, in most cases the NPV function turned out to be concave with a unique maximizing value. The second derivative of the NPV is provided in the Appendix for the special case of p(x) =p (constant). The sign of this expression
236
NPV PROFILE
NPV" (x) must be negative in order for the solution to be a local maximum.
Numerical examples
Linear Cost and Revenue Rates, T known NPV (x S10"9) 3.0
For the purposes of the numerical examples, we have used the same parameter values as in Krinsky and Parlar. This will enable us to identify the impact of making the discount rate a function of capacity on the NPV profile and on the optimal capacity level. We, thus, assume that one unit of capacity corresponds to a block of l 0,000 cars. It is also assumed that the lead time L(x), cost rate c(t) and revenue rate r(t) are all linear functions, i.e. L ( x ) = L x , c(t)=ct and r(t) = rt where L, c and r are given constants and the salvage value is zero. The discount rate p(x)=(5+8°°lX)/lO0 (e.g., when x=O, p=0.05; when x = 100, p=0.13). Initially, the following parameters values are employed: L=0.06 years/unit, c = $ 1 0 0 × 104/unit, r=$90×104/unit, and T = 1 5 years. Since L (x) ~ T the permissible values of x are limited to the [0, T/L ] interval. With the additional assumption of zero salvage value, the revenue and cost functions are obtained as R(X)=
[ (rx--re(Xptx)L-p(x)r)x - r e ( ' P ( x ) L - P ( x ) T)
( 5)
×xp(x) T+ rxep(x)L ) /p(x) 2] X e-O(x)Lx C( X ) "- [ ( - c x + ceXptX)Lx
(6)
- c x 2 p ( x ) L ) /p( x ) 2le -p(x)Lx The profit function N P V ( x ) = R ( x ) - C ( x ) with the above parameter values is plotted in Fig. 1. Optimization of NPV(x) results in x* =60.511 units (i.e. a capacity of 605,115 cars) and NPV of$2.12 X 109. One should note that this capacity is lower than the one obtained when p, the real discount rate, was assumed to be constant at p=0.05 (i.e., p' (x)=O). In this case x* would be equal to 875,850 cars with an optimal expected NPV of $3.53× 109. (See Krinsky and Parlar [ 5 ]. ) When the risk adjusted discount rate is assumed to be constant, an increase in the capacity level x reduces NPV via an increase in the construction costs and an increase in the lead
2.S
2.0
1.5
1.0
'111.0
'' ~ - o
2o
40
8o 8o loo Capacity (x 10,000)
1~o
14o
18o
Fig. 1. NPV profile. Linear cost and revenue rates, T known.
time (i.e., delay production start). Increase capacity, however, will also increase the net cash inflows generated once production starts. Making the ~isk adjusted discount rate an increasing function of capacity introduces an additional factor which supresses the NPV. The net benefit of an additional unit of capacity is discounted using a higher rate, thus, reducing the project's NPV.
Sensitivity analysis We first vary each ofp(x), L and Taround the initially assumed values and the results in Fig. 2 are obtained for the corresponding values of x* and NPV*. Note that increased valdes of-r. (end of product life cycle) sharply increase the corresponding values of the optimal NPV after T= 15. On the other hand increases in NFV* a :e not very pronounced when Tincreases f?om lower levels (between 5 to 10). Unlike the case where p is con-
237 unit capacity, when T is fixed, would result in lower profits due to shorter remaining time for profit generation. Further, as a result of the discount rate being a function of capacity, those profits are to be discounted using a high discount rate. Similar to Krinsky and Parlar [ 5 ] we also obtain that when varying L, Jr* and NPV* were found to be related linearly for different values of lead time L. The sensitivity of the NPV and x* to changes in the risk adjusted discount rate p is examined by changing the shift parameter o~ in p(x)=(Ol+8°°L~)/lO0. One can see that the NPV is very sensitivity to increased interest rates while capacity is less affected by the same changes. When the firm cost of capital increases giving up resources now is too expensive and is not compensated by the decline in capacity, resulting in a lower NPV. As in Fig. 2, one can observe in Fig. 3 the rate of change in capacity and NPV when the c and r parameters are changed. Although increasing c
Sensitivity Analysis for rho, L. and T o f optima| x and P (1" known) (rho = (0+8 ^ (0.01x))/100)
Movements
NPV(xSl0^S) 4-
T=23
a=2
f
=0.04
1 -
20
3o
4o so so Capacity (x 10,000) T L rho
70
Sensitivity Analysis for c and r
so
Fig. 2. Sensitivity analysis for p,L and T. Movements of optimal x and P; T known, p = ( a + 8 °°L~) / 100.
Movements
o f o p t i m a l x and P (T k n o w n )
NPV (x $10 ^9) s " r=175E+04
stant, the increase in the NPV when Tis increased (when p=p(x)) is not accompanied by an increased capacity x. In fact, capacity level declines for very high T values. This is quite obvious. An increase in T, lengthens the period during which net benefits can be generated while increased capacity increases the discount rate used in capitalizing these benefits. At a point, the increased benefits due to high T are overshadowed by the cost associated with a higher discount rate. Thus, x is kept almost constant (for T> 15) resulting in net benefits, in terms of increased NPV, for higher T values. In the constant p case, the only variables that supresses capacity are the after tax construction costs, and the l,:adtime. Changing the lead-time parameter (i.e. increasing it) reduces both the capacity and the NPV* (ceteris paribus). This is probably an expected result since longer construction times per
~
E+04
C:2|OE~L/
r-25E+04
40
4s
so
ss
so
ss
?o
Capacity (x 10,000) c
r
Fig. 3. Sensitivity analysis for c and r. Movements of optimal x and P; l" known.
238 reduces both x* and NPV* as expected (see Krinsky and Parlar [5]), and increasing r increases x* and NPV*, the rate of increase in the optimal profit turns out to be very substantial when r is increased. A result such as this may indicate that increasing net cash flows may influence the profit more than reducing construction cost. In the following section, the same problem is analyzed, assuming that the end of product life T is a random variable.
Similar expressions can be derived for the salvage value S, to obtain
$e-P{X'tSo(t)x, if t>~L(x) S(x)=(O, ift<~L(x)
(9)
and GO
E[S(x)
]=~e-P{"~tSo(t)xf(t)dt.
(10)
L
3. The model with random T
In the previous section, the end of the product life-time Twas assumed to be known in advance. In most situations, however, this assumption may not be valid. In realty, the consequences of technological advances, competitive pricing structure, unexpected changes in consumer tastes and preferences, and the impact of complementary products virtually assure that Tis a random variable that cannot be estimated with certainty. While FMS are able to respond to changes in consumer tastes and competitive pricing structure, it might not be able to react to the introduction of new technologies. The new flexible production facilities introduced by GM in its new plants will be of limited use if composite materials and plastics, are to replace steel in car manufacturing. Assume that the product lifetime T is a nonnegative random variable with density j r ( t ) and c.d.f. Fr(t) = Prob{ T~
ift~L(x) Namely, if the product life ends after the plant is built, the actual realized PV of revenues is the integral given above. If the product life ends prior to beginning of production, revenues will be equal to zero. One can, therefore, obtain the following expression as the expected value of/~,
LL
Of course, once capacity level is determined, the construction costs C(x) will not be affected by the realized value of T. Therefore, L(x)
C(x)= i
e-°~x~"xc(u)du
(11)
0
and the expected NPV to be equal to:
E[NfiV(x) ] = E [ / ~ ( x ) ] ....C(x)+E[S(x)]
(12)
which has to be maxi'~aized with respect to capacity level x. An examin,tion of 1 m necessary condition indicates that qualitati .'e analysis of eqn. (12) would be almost impossible to conduct. In fact, for complicated density functions for T, numerical analysis seems to be the only possibility for findings the optimal solution for ( 12 ). In some simple cases such as the uniformly distributed T one can explicitly integrate all the functions and find the solution suing a numerical technique; provided that the lead time, revenue-rate and cost-rate functions have simple structures. To provide a comparison between the basic model and the random T model and in order to explore the impact of making the discount rate a function of capacity, p (x), the same parameters values as in Krinsky and Parlar [ 5 ] are used. It is assumed that the random variable T is uniformly distributed between 0 and 30 giving E ( T ) = 15. Asbefore, L(x),r(t) and c(t) areassumed to be linear functions with the same parameter values. The values for the initial prob-
239
Expected NPV
Sensitivity Analysis for rho, L & Ymax
Linear Cost and Revenue Rates, T random
Movements of optimal x and P (T random) ( r h o : ( 0 + 8 ^ (0.01x))/1G0)
5xpected PV of Profit (x $10 ^ 9)
NPV of Profit (x $ 1 0 " 9 ) 2.5
s
,
2.0
T=60 4 1.S
i 7 0=2 1.0 7i I/
! O.S
L= 0.(~,
2 0.0
~.S
1 •
\ -1.0 a=24
3O 0
20
40
60 eo 100 Capacity (x 10,000)
120
140
160
rho
Fig. 4. Expected NPV. Linear cost and revenue rates, T random.
lem a r e : p ( x ) = ( 5 + 8 ° ° l X ) / l O 0 , L = 0 . 6 years/ unit, c = $ 1 0 0 × 104/unit, r = $ 9 0 × 104/unit, and T uniform over [ 0,Tmax ] with Tmax-" 30 and E ( T ) = 15 years. Of course, one unit still corresponds to a block of 10,000 cars. It is again assumed that the salvage value is equal to zero. The expected revenue and cost functions are equal to: E [ R ( x ) 1= [ (-2p(x)Lx2r
+p(x)xrTmax-2xr + e (p(.,-,L,-p t.,", rma~)p(x)XrTmax
+ 2e(PtX)L"-p(.")rma~)xr+p(x)2Lx2rTma~ -p(x)2L2x3r)/(p(x)3T~a~)]e -ptx)L" (13)
C(x)= [ (-cx+ce"("~Lx -cx2p(x)L)/p(x)E]e -p(')L"
40 5O Capacity (x 10,000)
(14)
Figure 4 depicts the expected NPV function E [ N P V ( x ) ] = E [ / ~ ( x ) ] - C ( x ) , for 0~
L
6O
Trnax
Fig. 5. Sensitivity analysis for p,L and T m a X. Movements of optimal x and P; T known, p = ( a + 8 °°r~ ) / 100.
E ( T ) = 15 years, optimization of E [ N P V ( x ) ] results in x*=57.834 units (i.e. a capacity of 578.340 cars) and E*[NPV(x) ] =$2.08X l09. Namely, the random T results in a lower optimal plant size and a lower NPV than the deterministic case. This is in sharp contrast to IGinsky and Parlar model with the constant discount rate in which the random case provided a higher optimal plant size as well as a higher NPV. The decrease in, both, the optimal capacity and optimal expected profit with random (uniform) Tcan be explained as follows. The chances of actual value of T being less than 15 years or greater than 15 years are equal at 50%. When T is realized between 15 and 30, the realized PV of revenues is lower than the case when T was a deterministic 15 y~ars Whereas when f is a realized between 0 and 15, chances of no generation of revenue increases, but since the model does not decrease the value of optimal x too much compared to the de-
70
240 terministic Tcase, likelihood of zero revenue does not affect the objective function very much, especially when compared to the effects of having T between 15 and 30 years. When the firm's cost of capital is high, giving up dollars now is too expensive and is not compensated for by lengthening the collection period in the future. Some sensitivity analyses on the same groups ofnonmonetary and monetary parameters is carried out. This time since T is a random variable, the effects of the upper bound Tma~ of the uniform distribution on the optimal capacity x a~.d the expected NPV, E [ NPV (x) ], are analyzed indicating that the randomness of Thad an adverse effect on these variables. These results are provided in Figure 5.
4. Summary and suggestions for future research In this paper a model which explicitly recognizes the relationships between the plant capacity, the lead time for building the plant, and the discount rate used in evaluating the project, is developed. It is shown that the introduction of a discount rate which is a function of capacity, substantially alters the results obtained in a previous study where the discount rate is assumed to be constant. In particular, since the net benefits of an additional unit of capacity is discounted using a higher rate, the NPV is supressed. An extensive sensitivity analyses on the nonmonetary and monetary parameters and their effect on the optimal plant size vs. optimal NPV is provided. The case of random "end of the product life" problem is also discussed. Our sensitivity analyses on the same nonmonetary and monetary parameters revealed that with the uniform distribution assumption for the random variable, both the optimal plant size and the optimal expected NPV took smaller values than they had in the deterministic case. Again these results differ from the constant discount rate case where the introduction of a random "end of the product life" resulted in higher value for both the optimal expected NPV and the optimal plant size. The resuits obtained highlight the important role that the discount rate is playing in any capital budgeting decision. The failure to consider the relationship between the discount rate and the plant
capacity in capital budgeting decisions may mislead managers with regard to the optimal plant capacity and the profitability of the project. Further, the numerical analysis makes it readily apparent that financial managers may be ignoring a significant risk factor associated with capital budgeting by assuming that the project's life is certain. A possible extension of the model presented would be to use option pricing in order to formalize the relationship between capacity and capital budgeting decision. This will allow the relaxation of the current model's assumptions with regard to the nature of the various cash flows. In particular, the after tax real net cash flow rate per unit of capacity, r(t), and the after tax real construction rate per unit of capacity, c(t), can be described by stochastic process. The use of an option model might also provide a close form solution to the problem so that a simulation will no longer be required.
Acknowledgments The author would like to thank the two anonymous referees for their constructive comments, Mahmut Parlar for many helpful and sometimes lengthy discussions, and the Nat ional Science and Engineering Research Council of Canada for financial support ander grant OGP0042195.
References 1 Merrow, E., Phillips, K. and Myers, C., 1981. Understanding Cost Growth and Performance in Pioneer Process Plants, California Rand Corp., Santa Barbara. 2 Ross, M., 1986. Capital budgeting practices of twelve large manufacturers. Finan. Manage., 15(4): i 5-22. 3 Fisher, A.B., 1985. Behind the hype at GM's Saturn. Fortune, 112 ( 11 ): 34-49. 4 Fuller, J.D. and Y. Gerchak, 1989. Risk-aversion, output rate and plant size: theory and application to tar-sands oil plants. Can. J. Econ., 22: 188-197. 5 Krinsky, I. and M. Parlar, 1989. Optimal plant size with variable cost and revenue streams, Eur. J. Oper. Res., 43: 78-87. 6 Wagle, B., 1967. A statistical analysis of risk in capital investment projects, Oper. Res. Q., 18( I ): 13-33. 7 Bey, R.P., 1980. Calculating means and variances of NPVs when the life of the project is uncertain. J. Finan. Res., Ill(2): 139-152. 8 Bey, R.P., 1983. Capilal budgeting decisions when cash
241 flows and project lives are stochastic and dependent. J. Finan. Res.. VI(3): 175-185. 9 Brealy,R.A. and S.C. Myers, 1988. Principles of Corporate Finance, 3rd ed., McGraw-Hill,New York.
Appendix Analysis when p(x) = p= constant Using Leibniz's rule one can obtain that
R" (x)= { - x r ( L )L" +pxr(L ) (L' )2
T
R'(x)=
enue" interpretation and we obtain C' (x) = R' (x) + S' (x) as the necessary condition for NPV maximization. Note that due to the complicated nature of ( A . l ) - ( A . 3 ) , it is impossible to obtain a close form solution for NPV' ( x ) = 0 . The second derivatives of the individual components of NPV (x) are:
- [ x r ' , ( L ) - 2 r ( L ) ]L'}e -pL
~ e-m'r(u)du-xL'e-'Lr(L)
C" (x) = { [xL" - p x ( L' )2 + 2L' ]c(L)
L(x)
+ xL' C' x' L }e -pL
I
-- x ~ p' ur(u)e-P"du
(A.I)
L(x )
where
L(x)
r .',.(L) = ( d r / d L ) (dL/dx)
C' ( x ) = ~ e-puc(u)du+xL'e-pLc(L)
and c',(L ) = ( d c / d L ) (dL/dx).
0
These give
L(x)
--x ~ p'uc(u)e-P"du
S"(x)=0
(A.2)
NPV"(x)=R"(x)-C"(x)
0
S' (x)=so(T)e-Pr-Txso(T)p'e -p'
(A.3)
Clearly, NPV' (x)=R' ( x ) - C ' (x)+S' (x)=O leads to the usual "marginal cost = marginal rev-
whose sign must be checked at the stationary point which solves NPV' ( x ) = 0 . R" (x), C" (x), and S" (x) for the general case p ( x ) might be obtained directly from the authors.
233
Elsevier
Capital budgeting and plant capacity ltzhak Krinsky Finance and Business Economics, Faculty of Business~ McMaster University, Hamilton. Ontario. Canada L8S 4M4 (Received 18 June, 1990; accepted in revised form 20 November 1990)
Abstract In this paper a model which explicitly recognizes the relationships between the plant capacity, the lead time for building the plant, and the discount rate used in evaluating the project, is developed. It is shown that the introduction of a discount rate which is a function of capacity, substantially alters the results obtained in a previous study where the discount rate is assumed to be constant. In particular, since the net benefits of an additionel unil of capacity is dicounted uz,ing a higher rate, the NPV is suppressed. An extensive sensitivity analyses on the nonmonetaq,' and monetary parameters and their effect on the optimal plant size vs. optima! NPV is provided.
|. Introduction As a result of the high degree of uncertainty, many projects in recent years are characterized by capital expenditure overruns and poor performance. Research reports often verify the general impression that estimates of their performance, tim,~tables, and costs are extraordinarily accident-prone, particularly when new techno!ogies are involved (see, for example, Merrow, Phillips, and Myers [1 ] ). If a company is contemplating a major investment project that involves a new technology, market, or a new size ~o the company, serious overruns are a real possibility. Frequentb ~~ost overruns are a result of an optimistic forecast of the plant's construction time and an unrealistic assessment of the difficulties one might face when exploring new territories. Ross [2 ], in reviewing the capital budgeting process of twelve large manufacturers, points out that even in cases where sensitivity analysis for potential risk was not avoided, the projects' "benefits and economic lives tended to be less than predicted because of unanticipated changes in production rates and technology" (p. 18 ). He adds that these insights were, usually, not incorporated in a formal fashion in the financial analysis.
0167-188X/91/$03.50 © 199 I--Elsevier Science Publishers B.V.
An obvious example is General Motors (GM). In its quest to secure a place in the subcompact car market, GM decided at the beginning of 1985 to spend 3.5 billion dollars to build its Saturn plant, about six times the 600 million dollars it spent to construct its Hamtramch assembly plant in Detroit which is considered one of the world's most technologically advanced. At an original production capacity of half a million cars a year, it was expected that the Saturn plant could build a car in 40 hours of labour (compared to a total of 100 labour hours that Japanese car makers now put). The plant was to be built with an automatic guided vehicle (AGV) system. GM also entered into a joint venture with the Japanese robot maker, Fanuc, to develop various machines to save labour and boost product quality. GM chairman has stated that "Saturn is the key to GM's long term competitiveness, survival, and success as a domestic producer." [ Fisher [ 3 ] p. 8]. The Saturn car project of GM has come a long way since 1985. In a shareholders meeting held in May 1987 GM announced that, due to changes in market conditions, the Saturn plant will not be able to compete with South Korean and other foreign manufacturers. It has also slashed production plans to 250,000 cars a year from
234 500,000, and the labour force to 3,000 from the 6,000 originally announced. In fact, the plant is not scheduled to start production before the end of 1990 and one might question the profitability of the entire project. In this paper, the problem faced by a decision maker described in the previous paragraphs is analysed. Specifically, it deals with the optimal plant capacity in a changing environment where the product's life cycle ends after T periods. Further, the relationships between the plant capacity, the lead time for building the plant, and the discount rate used in evaluating the project, are explicitly recognized. A realistic framework in which the impact of stochastic projects lives on the expected net present value and plant capacity, is also provided. While the model deals with capacity, it is also applicable to cases where the lead time is a function of complexity rather than capacity (i.e., cases where advanced manufacturing technologies are introduced). In what follows, a simple NPV maximization model for a company assumed to be a price taker is developed in Section II. The problem is to choose an optimal plant size so as to maximize the NPV. As sucb, our model is in the spirit of Fuller and Gerchak [ 4 ] and, especially, Krinsky and Parlar [5 ]. The major differences between the results provided here and those in previous studies is the fact that the discoun'., rate is assumed to be a function of capacity. Assuming T as a known parameter and the discount rate p dependent of capacity, we analyze the concavity properties of the NPV. For a wide range of parameters we have tested, NPV was concave. The results are illustrated with several numerical examples. The sensitivity of the optimal solution to various monetary and nonmonetary parameters is also investigated. In Section 3, the end of life cycle T is assumed to be a nonnegative random variable with a known probability density function. The incorporation of a stochastic - - not new. The work by Wagle [6 ], and especially Bey [7,8 ], among others, addresses some aspects of the stochastic project life problem. None, however, deals with the problem within a model which explicitly reco ognizes the relationship between the plant capacity and the lead ;ime to build the plant. The very
complicated nature of the resulting expected NPV function makes it difficult to analyze its qualitative properties. Nevertheless, numerical experiments indicate that this function as the one with deterministic T is concave, at least for the practical range of parameter assumed. The paper is concluded with a summary of findings and a discussion on the implications of the results. 2. The basic model
The basic model is similar to the one introduced by Krinsky and Parlar [ 5 ] in which p is assumed to be a parameter independent of capacity. Let x be the plant size, measured in production capacity per unit of time. Here, x is the decision variable to be chosen at time t = 0. Define the following. L (x) = Construction lead time of plant of size x. Here L ( 0 ) =0, L' (x) >0, L" (x)>~0 and L (x) ~
235 does not serve the purpose of the paper. Krinsky and Parlar [5] also achieve risk adjustment in a similar model by utilizing a riskadjusted discount rate. The risk level in their model, however, is not a function of plant capacity - it is constant over all capacity levels x. The risk-adjusted discount rate in this model, is assumed to be an increasing function of capacity. This is so because all companies with non-zero beta would find each marginal unit to be sold to be less likely, except in better state outcomes. In other words, the systematic risk of the marginal unit increases with capacity. When studying various energy-saving projects, Ross [2] states that contingencies such as reduced production because of weak sales or changes in the production process are often ignored when the projects are evaluated. He also observes that smaller projects usually had higher hurdle rates despite the fact that there was no reason "that would justify subjecting them to higher hurdle rates" (p. 20). As a result, smaller projects were discriminated against. As shown below, making the discount rate a function of capacity has a major impact on the optimal capacity level. Given the above definitions, the present value (PV) at time t = 0 of the after tax net cash flows is equal to
S(x) = e-~"" ~/-~-"~So( T ) x e
-pc~~ t T - Lcx) 1
=so(T)xe -p~-'~r
(3)
is the PV of the salvage value when the plant is salvaged at time T. Combining ( l ) - ( 3 ) gives the NPV to be equal to NPV(x)
=R(x) -C(x) + S(x)
(4)
which is to be maximized with respect to capacity x. Note that NPV (0) =0, i.e. if the capacity is at zero level, then the NPV is also zero. Next, at the other end point x-x~ =L-~ (T) one can obtain R(xt)=0 "d"
C(x~ ) = Se -''~ ' ~"xl c( u )du o
and
S(xl)
=So( T)xl e -pI'~T
so that 1"
NPV(xl) = -
[
fe-m")'c(u)du
a¢
o
--So( T)e-PC~)r]Xl < 0 since
7"
T
R(x) =e-p'''~Lc'' ~ e-PC"~t'-L{x~]xr(u)du
f
L(x)
e-"~-""c ( u )du> So( T)e -ptx~r
o
T
= ~ e-PC"'"xr(u)du
(l)
L(x)
Also, L(.v)
C ( x ) = e -p~')L~'~
f et'~'~tL~"}-UJxc(u)du o
L(x)
= ~ e-P''}"xc(u)du
(2)
o
is the PV of construction costs incurred between 0 and L (x). Further,
i.e the PV of construction cost of one unit capacity for T time units is greater than the PV of salvage value of that unit capaci.ty due to the fact that no revem~es will be generated. Therefore, NPV (0) = 0 and NPV (xt) < 0 at the two feasible end points of the decision variable. We were unable to prove the concavity of NPV(x) for general p(x) due to the extreme complexity of the second derivative NPV" (x). Nevertheless, in the numerical examples we have conducted, in most cases the NPV function turned out to be concave with a unique maximizing value. The second derivative of the NPV is provided in the Appendix for the special case of p(x) =p (constant). The sign of this expression
236
NPV PROFILE
NPV" (x) must be negative in order for the solution to be a local maximum.
Numerical examples
Linear Cost and Revenue Rates, T known NPV (x S10"9) 3.0
For the purposes of the numerical examples, we have used the same parameter values as in Krinsky and Parlar. This will enable us to identify the impact of making the discount rate a function of capacity on the NPV profile and on the optimal capacity level. We, thus, assume that one unit of capacity corresponds to a block of l 0,000 cars. It is also assumed that the lead time L(x), cost rate c(t) and revenue rate r(t) are all linear functions, i.e. L ( x ) = L x , c(t)=ct and r(t) = rt where L, c and r are given constants and the salvage value is zero. The discount rate p(x)=(5+8°°lX)/lO0 (e.g., when x=O, p=0.05; when x = 100, p=0.13). Initially, the following parameters values are employed: L=0.06 years/unit, c = $ 1 0 0 × 104/unit, r=$90×104/unit, and T = 1 5 years. Since L (x) ~ T the permissible values of x are limited to the [0, T/L ] interval. With the additional assumption of zero salvage value, the revenue and cost functions are obtained as R(X)=
[ (rx--re(Xptx)L-p(x)r)x - r e ( ' P ( x ) L - P ( x ) T)
( 5)
×xp(x) T+ rxep(x)L ) /p(x) 2] X e-O(x)Lx C( X ) "- [ ( - c x + ceXptX)Lx
(6)
- c x 2 p ( x ) L ) /p( x ) 2le -p(x)Lx The profit function N P V ( x ) = R ( x ) - C ( x ) with the above parameter values is plotted in Fig. 1. Optimization of NPV(x) results in x* =60.511 units (i.e. a capacity of 605,115 cars) and NPV of$2.12 X 109. One should note that this capacity is lower than the one obtained when p, the real discount rate, was assumed to be constant at p=0.05 (i.e., p' (x)=O). In this case x* would be equal to 875,850 cars with an optimal expected NPV of $3.53× 109. (See Krinsky and Parlar [ 5 ]. ) When the risk adjusted discount rate is assumed to be constant, an increase in the capacity level x reduces NPV via an increase in the construction costs and an increase in the lead
2.S
2.0
1.5
1.0
'111.0
'' ~ - o
2o
40
8o 8o loo Capacity (x 10,000)
1~o
14o
18o
Fig. 1. NPV profile. Linear cost and revenue rates, T known.
time (i.e., delay production start). Increase capacity, however, will also increase the net cash inflows generated once production starts. Making the ~isk adjusted discount rate an increasing function of capacity introduces an additional factor which supresses the NPV. The net benefit of an additional unit of capacity is discounted using a higher rate, thus, reducing the project's NPV.
Sensitivity analysis We first vary each ofp(x), L and Taround the initially assumed values and the results in Fig. 2 are obtained for the corresponding values of x* and NPV*. Note that increased valdes of-r. (end of product life cycle) sharply increase the corresponding values of the optimal NPV after T= 15. On the other hand increases in NFV* a :e not very pronounced when Tincreases f?om lower levels (between 5 to 10). Unlike the case where p is con-
237 unit capacity, when T is fixed, would result in lower profits due to shorter remaining time for profit generation. Further, as a result of the discount rate being a function of capacity, those profits are to be discounted using a high discount rate. Similar to Krinsky and Parlar [ 5 ] we also obtain that when varying L, Jr* and NPV* were found to be related linearly for different values of lead time L. The sensitivity of the NPV and x* to changes in the risk adjusted discount rate p is examined by changing the shift parameter o~ in p(x)=(Ol+8°°L~)/lO0. One can see that the NPV is very sensitivity to increased interest rates while capacity is less affected by the same changes. When the firm cost of capital increases giving up resources now is too expensive and is not compensated by the decline in capacity, resulting in a lower NPV. As in Fig. 2, one can observe in Fig. 3 the rate of change in capacity and NPV when the c and r parameters are changed. Although increasing c
Sensitivity Analysis for rho, L. and T o f optima| x and P (1" known) (rho = (0+8 ^ (0.01x))/100)
Movements
NPV(xSl0^S) 4-
T=23
a=2
f
=0.04
1 -
20
3o
4o so so Capacity (x 10,000) T L rho
70
Sensitivity Analysis for c and r
so
Fig. 2. Sensitivity analysis for p,L and T. Movements of optimal x and P; T known, p = ( a + 8 °°L~) / 100.
Movements
o f o p t i m a l x and P (T k n o w n )
NPV (x $10 ^9) s " r=175E+04
stant, the increase in the NPV when Tis increased (when p=p(x)) is not accompanied by an increased capacity x. In fact, capacity level declines for very high T values. This is quite obvious. An increase in T, lengthens the period during which net benefits can be generated while increased capacity increases the discount rate used in capitalizing these benefits. At a point, the increased benefits due to high T are overshadowed by the cost associated with a higher discount rate. Thus, x is kept almost constant (for T> 15) resulting in net benefits, in terms of increased NPV, for higher T values. In the constant p case, the only variables that supresses capacity are the after tax construction costs, and the l,:adtime. Changing the lead-time parameter (i.e. increasing it) reduces both the capacity and the NPV* (ceteris paribus). This is probably an expected result since longer construction times per
~
E+04
C:2|OE~L/
r-25E+04
40
4s
so
ss
so
ss
?o
Capacity (x 10,000) c
r
Fig. 3. Sensitivity analysis for c and r. Movements of optimal x and P; l" known.
238 reduces both x* and NPV* as expected (see Krinsky and Parlar [5]), and increasing r increases x* and NPV*, the rate of increase in the optimal profit turns out to be very substantial when r is increased. A result such as this may indicate that increasing net cash flows may influence the profit more than reducing construction cost. In the following section, the same problem is analyzed, assuming that the end of product life T is a random variable.
Similar expressions can be derived for the salvage value S, to obtain
$e-P{X'tSo(t)x, if t>~L(x) S(x)=(O, ift<~L(x)
(9)
and GO
E[S(x)
]=~e-P{"~tSo(t)xf(t)dt.
(10)
L
3. The model with random T
In the previous section, the end of the product life-time Twas assumed to be known in advance. In most situations, however, this assumption may not be valid. In realty, the consequences of technological advances, competitive pricing structure, unexpected changes in consumer tastes and preferences, and the impact of complementary products virtually assure that Tis a random variable that cannot be estimated with certainty. While FMS are able to respond to changes in consumer tastes and competitive pricing structure, it might not be able to react to the introduction of new technologies. The new flexible production facilities introduced by GM in its new plants will be of limited use if composite materials and plastics, are to replace steel in car manufacturing. Assume that the product lifetime T is a nonnegative random variable with density j r ( t ) and c.d.f. Fr(t) = Prob{ T~
ift~L(x) Namely, if the product life ends after the plant is built, the actual realized PV of revenues is the integral given above. If the product life ends prior to beginning of production, revenues will be equal to zero. One can, therefore, obtain the following expression as the expected value of/~,
LL
Of course, once capacity level is determined, the construction costs C(x) will not be affected by the realized value of T. Therefore, L(x)
C(x)= i
e-°~x~"xc(u)du
(11)
0
and the expected NPV to be equal to:
E[NfiV(x) ] = E [ / ~ ( x ) ] ....C(x)+E[S(x)]
(12)
which has to be maxi'~aized with respect to capacity level x. An examin,tion of 1 m necessary condition indicates that qualitati .'e analysis of eqn. (12) would be almost impossible to conduct. In fact, for complicated density functions for T, numerical analysis seems to be the only possibility for findings the optimal solution for ( 12 ). In some simple cases such as the uniformly distributed T one can explicitly integrate all the functions and find the solution suing a numerical technique; provided that the lead time, revenue-rate and cost-rate functions have simple structures. To provide a comparison between the basic model and the random T model and in order to explore the impact of making the discount rate a function of capacity, p (x), the same parameters values as in Krinsky and Parlar [ 5 ] are used. It is assumed that the random variable T is uniformly distributed between 0 and 30 giving E ( T ) = 15. Asbefore, L(x),r(t) and c(t) areassumed to be linear functions with the same parameter values. The values for the initial prob-
239
Expected NPV
Sensitivity Analysis for rho, L & Ymax
Linear Cost and Revenue Rates, T random
Movements of optimal x and P (T random) ( r h o : ( 0 + 8 ^ (0.01x))/1G0)
5xpected PV of Profit (x $10 ^ 9)
NPV of Profit (x $ 1 0 " 9 ) 2.5
s
,
2.0
T=60 4 1.S
i 7 0=2 1.0 7i I/
! O.S
L= 0.(~,
2 0.0
~.S
1 •
\ -1.0 a=24
3O 0
20
40
60 eo 100 Capacity (x 10,000)
120
140
160
rho
Fig. 4. Expected NPV. Linear cost and revenue rates, T random.
lem a r e : p ( x ) = ( 5 + 8 ° ° l X ) / l O 0 , L = 0 . 6 years/ unit, c = $ 1 0 0 × 104/unit, r = $ 9 0 × 104/unit, and T uniform over [ 0,Tmax ] with Tmax-" 30 and E ( T ) = 15 years. Of course, one unit still corresponds to a block of 10,000 cars. It is again assumed that the salvage value is equal to zero. The expected revenue and cost functions are equal to: E [ R ( x ) 1= [ (-2p(x)Lx2r
+p(x)xrTmax-2xr + e (p(.,-,L,-p t.,", rma~)p(x)XrTmax
+ 2e(PtX)L"-p(.")rma~)xr+p(x)2Lx2rTma~ -p(x)2L2x3r)/(p(x)3T~a~)]e -ptx)L" (13)
C(x)= [ (-cx+ce"("~Lx -cx2p(x)L)/p(x)E]e -p(')L"
40 5O Capacity (x 10,000)
(14)
Figure 4 depicts the expected NPV function E [ N P V ( x ) ] = E [ / ~ ( x ) ] - C ( x ) , for 0~
L
6O
Trnax
Fig. 5. Sensitivity analysis for p,L and T m a X. Movements of optimal x and P; T known, p = ( a + 8 °°r~ ) / 100.
E ( T ) = 15 years, optimization of E [ N P V ( x ) ] results in x*=57.834 units (i.e. a capacity of 578.340 cars) and E*[NPV(x) ] =$2.08X l09. Namely, the random T results in a lower optimal plant size and a lower NPV than the deterministic case. This is in sharp contrast to IGinsky and Parlar model with the constant discount rate in which the random case provided a higher optimal plant size as well as a higher NPV. The decrease in, both, the optimal capacity and optimal expected profit with random (uniform) Tcan be explained as follows. The chances of actual value of T being less than 15 years or greater than 15 years are equal at 50%. When T is realized between 15 and 30, the realized PV of revenues is lower than the case when T was a deterministic 15 y~ars Whereas when f is a realized between 0 and 15, chances of no generation of revenue increases, but since the model does not decrease the value of optimal x too much compared to the de-
70
240 terministic Tcase, likelihood of zero revenue does not affect the objective function very much, especially when compared to the effects of having T between 15 and 30 years. When the firm's cost of capital is high, giving up dollars now is too expensive and is not compensated for by lengthening the collection period in the future. Some sensitivity analyses on the same groups ofnonmonetary and monetary parameters is carried out. This time since T is a random variable, the effects of the upper bound Tma~ of the uniform distribution on the optimal capacity x a~.d the expected NPV, E [ NPV (x) ], are analyzed indicating that the randomness of Thad an adverse effect on these variables. These results are provided in Figure 5.
4. Summary and suggestions for future research In this paper a model which explicitly recognizes the relationships between the plant capacity, the lead time for building the plant, and the discount rate used in evaluating the project, is developed. It is shown that the introduction of a discount rate which is a function of capacity, substantially alters the results obtained in a previous study where the discount rate is assumed to be constant. In particular, since the net benefits of an additional unit of capacity is discounted using a higher rate, the NPV is supressed. An extensive sensitivity analyses on the nonmonetary and monetary parameters and their effect on the optimal plant size vs. optimal NPV is provided. The case of random "end of the product life" problem is also discussed. Our sensitivity analyses on the same nonmonetary and monetary parameters revealed that with the uniform distribution assumption for the random variable, both the optimal plant size and the optimal expected NPV took smaller values than they had in the deterministic case. Again these results differ from the constant discount rate case where the introduction of a random "end of the product life" resulted in higher value for both the optimal expected NPV and the optimal plant size. The resuits obtained highlight the important role that the discount rate is playing in any capital budgeting decision. The failure to consider the relationship between the discount rate and the plant
capacity in capital budgeting decisions may mislead managers with regard to the optimal plant capacity and the profitability of the project. Further, the numerical analysis makes it readily apparent that financial managers may be ignoring a significant risk factor associated with capital budgeting by assuming that the project's life is certain. A possible extension of the model presented would be to use option pricing in order to formalize the relationship between capacity and capital budgeting decision. This will allow the relaxation of the current model's assumptions with regard to the nature of the various cash flows. In particular, the after tax real net cash flow rate per unit of capacity, r(t), and the after tax real construction rate per unit of capacity, c(t), can be described by stochastic process. The use of an option model might also provide a close form solution to the problem so that a simulation will no longer be required.
Acknowledgments The author would like to thank the two anonymous referees for their constructive comments, Mahmut Parlar for many helpful and sometimes lengthy discussions, and the Nat ional Science and Engineering Research Council of Canada for financial support ander grant OGP0042195.
References 1 Merrow, E., Phillips, K. and Myers, C., 1981. Understanding Cost Growth and Performance in Pioneer Process Plants, California Rand Corp., Santa Barbara. 2 Ross, M., 1986. Capital budgeting practices of twelve large manufacturers. Finan. Manage., 15(4): i 5-22. 3 Fisher, A.B., 1985. Behind the hype at GM's Saturn. Fortune, 112 ( 11 ): 34-49. 4 Fuller, J.D. and Y. Gerchak, 1989. Risk-aversion, output rate and plant size: theory and application to tar-sands oil plants. Can. J. Econ., 22: 188-197. 5 Krinsky, I. and M. Parlar, 1989. Optimal plant size with variable cost and revenue streams, Eur. J. Oper. Res., 43: 78-87. 6 Wagle, B., 1967. A statistical analysis of risk in capital investment projects, Oper. Res. Q., 18( I ): 13-33. 7 Bey, R.P., 1980. Calculating means and variances of NPVs when the life of the project is uncertain. J. Finan. Res., Ill(2): 139-152. 8 Bey, R.P., 1983. Capilal budgeting decisions when cash
241 flows and project lives are stochastic and dependent. J. Finan. Res.. VI(3): 175-185. 9 Brealy,R.A. and S.C. Myers, 1988. Principles of Corporate Finance, 3rd ed., McGraw-Hill,New York.
Appendix Analysis when p(x) = p= constant Using Leibniz's rule one can obtain that
R" (x)= { - x r ( L )L" +pxr(L ) (L' )2
T
R'(x)=
enue" interpretation and we obtain C' (x) = R' (x) + S' (x) as the necessary condition for NPV maximization. Note that due to the complicated nature of ( A . l ) - ( A . 3 ) , it is impossible to obtain a close form solution for NPV' ( x ) = 0 . The second derivatives of the individual components of NPV (x) are:
- [ x r ' , ( L ) - 2 r ( L ) ]L'}e -pL
~ e-m'r(u)du-xL'e-'Lr(L)
C" (x) = { [xL" - p x ( L' )2 + 2L' ]c(L)
L(x)
+ xL' C' x' L }e -pL
I
-- x ~ p' ur(u)e-P"du
(A.I)
L(x )
where
L(x)
r .',.(L) = ( d r / d L ) (dL/dx)
C' ( x ) = ~ e-puc(u)du+xL'e-pLc(L)
and c',(L ) = ( d c / d L ) (dL/dx).
0
These give
L(x)
--x ~ p'uc(u)e-P"du
S"(x)=0
(A.2)
NPV"(x)=R"(x)-C"(x)
0
S' (x)=so(T)e-Pr-Txso(T)p'e -p'
(A.3)
Clearly, NPV' (x)=R' ( x ) - C ' (x)+S' (x)=O leads to the usual "marginal cost = marginal rev-
whose sign must be checked at the stationary point which solves NPV' ( x ) = 0 . R" (x), C" (x), and S" (x) for the general case p ( x ) might be obtained directly from the authors.