Capital investment with abandonment options and uncertainty in project life using multiperiod CAPM

361

Capital investment with abandonment options and uncertainty in project life using multiperiod CAPM P.K. DE, D. ACHARYA and K.C. SAHU Idustriul Mmcrgentenr Centre. In&n Institute of Techdog,,,

Kharugpur-721302,

Id~u

Received March 1981 Revised June 1982

This paper presents a dynamic programming approach to evaluate the expected value and variance of the net present value of an investment while considering the abandonment options, both deterministic and probabilistic, in a multiperiod capital asset pricing model context. Uncertainty in project life is also considered in the analysis. A numerical example is given to illustrate the methodologies.

1. Introduction Robichek and Van Horne [6,7], Schwab and Lusztig [8], Bonini [2] and many others have emphasized the need of considering abandonment options in evaluating the economic desirability of a project. A project may have to be abandoned due to premature failure of a capital equipment. Also, it may be worthwhile to abandon a project before the receipts of all anticipated cash flows. if at some point in time the present worth of all the subsequent cash flows is less than the abandonment value of the project. Bonini [2] has presented a methodology to evaluate an investment assuming the cash flow generation to be a Markov process. His study employs an ordinary cash flow discounting procedure and considers deterministic project life only. His analysis includes the possibility of future abandonment options. It is intuitively convincing that the project evaluation becomes more realistic when the life of a project is considered as a stochastic variable. Though the necessity of considering the probabilistic nature of project life has been emphasized by Van Horne [9], Wagle [lo], De et al. [3]. Hertz [4], and others, hardly any attempt has been made so far to incorporate this important aspect in mathematical programming models for capital budgeting decisions. Furthermore, simultaneous consideration of abandonment options and uncertainty in project life has, in fact. not been noticed in the literature on project evaluation. Myers and Turnbull ([5]; hereafter M-T) have presented expressions to evaluate the current equilibrium value of an asset using the capital asset pricing model (CAPM) in a multiperiod context, assuming that the CAPM is applicable in each period. The CAPM has the ability to consider the interaction between project and market parameters [3,5]. Based on the multiperiod CAPM concepts of M-T, this paper presents a dynamic programming approach to evaluate the expected value and variance of the net present value (NPV) of an investment, taking account of future abandonment options. The analysis considers both deterministic and probabilistic abandonment value [2]. In the latter case, the abandonment value at a particular point of time is randomly distributed with a finite mean and variance. In case of deterministic abandonment, the abandonment value The authors wish to acknowledge the original paper.

North-Holland European Journal

of Operational

0377-22 17: 83 /$3.00

the helpful comments

Research

of the refrrees.

These comments

13 (1983) 361-368

(1 1983. Elsevier Science Publishers

B.V. (North-Holland)

have led to the substantial

improvement

of

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P.K. De et al. / Capital investment

uwzg multiperiod CAPM

is known with certainty. The study has further been extended to a more realistic situation by considering uncertainty in project life. Both independent and correlated cash flows have been considered. A numerical example is given to illustrate the models.

2. Transition probability for correlated cash flows Let %,, the cash flow at any time t (t = 1,2,. . . , T), be a normally distributed random variable and T be the life of the project. Under the usual Markovian assumption [2] of the cash flow time series, ,?,+, only depends on _J?,and is independent of the previous history. Further, the conditional distribution of 2, + , for given value of 2, is normally distributed. The mean (E( j,+, 1kf)) and the variance ( V( g,+, / il)) of this conditional probability distribution are

(2) where p is the coefficient of correlation between i,+, and 2,. Since, the dynamic programming models of the next sections only consider a finite number of discrete cash flow states in each period, we approximate the normally distributed cash flow by a discrete probability density function of (k,+ , I kr) distribution with d possible cash flow states. As the conditional is normal, the probability of transition ( p,,) from the cash flow state S, in period t to cash flow state S, in period t + 1 for a given p is thus easily determined. The transition probability matrix for p = 0.5 is shown in Table 1.

3. Project evaluation with certain project life Using the basic Myers-Turnbull model (Appendix A), we have derived dynamic programming relations to evaluate the NPV of a project considering the abandonment options in each period.

Table 1 Transition States in

t

2 3 4 5 6 7 8 9 10 11

probability

matrix,

recursive Let p,‘, be

p =0.5

Standard deviation from mean

States in t + 1: 1 2 Standard deviation 2.5 2.0

2.5 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 - 1.5 -2.0 -2.5

0.1241 0.0744 0.0416 0.0216 0.0104 0.0047 0.0019 0.0007 0.0002 0.0002 0.000 1

0.1578 0.1188 0.0825 0.0528 0.0321 0.0169 0.0085 0.0040 0.0016 0.0005 0.0002

3 4 from mean: 1.5 1.0

0.2181 0.1958 0.1578 0.1188 0.0825 0.0528 0.0312 0.0 169 0.0085 0.0040 0.0016

0.2181 0.2220 0.2181 0.1958 0.1578 0.1188 0.0825 0.0528 0.03 13 0.0169 0.0085

5

6

7

8

9

IO

I1

0.5

0.0

-0.5

~ 1.0

~ 1.5

~ 2.0

-2.5

0.1578 0.1958 0.2181 0.2220 0.2181 0.1958 0.1578 0.1188 0.0825 0.0528 0.0312

0.0825 0.1188 0.1578 0.1958 0.2181 0.2220 0.2181 0.1958 0.1578 0.1188 0.0825

0.03 12 0.0528 0.0825 0.1188 0.1571 0.1958 0.2181 0.2220 0.2181 0.1958 0.1578

0.0085 0.0169 0.03 13 0.0528 0.0825 0.1188 0.1578 0.1958 0.2181 0.2220 0.2181

0.0016 0.0040 0.0085 0.0169 0.0312 0.0528 0.0825 0.1188 0.1578 0.1958 0.2181

0.0002 0.0005 0.00 16 0.0040 0.0085 0.0169 0.0312 0.0528 0.0825 0.1188 0.1578

0.000 1 0.0002 0.0007 0.0007 0.0019 0.0047 0.0104 0.0216 0.0416 0.0744 0.1241

P. K. De Ed al. / Caprtal inwstment

usmg multiperiod

CAPM

363

the n times products of p,,; pO, be the probability for the initial cash flow state to bej in period 1; A,( S,) be the abandonment value at the end of period t, given the cash flow state S, during period t; and Q. be the initial investment. The dynamic programming recursive relations for NPV are (a) P,(S,)=A,(S,)

forallstatesS,att=T,

(3)

(b) P,(s,>

= Max[A,($);

B,($);

c,(s,)]-.

(4)

where

T--1

d

c,(x) = 2 T=I

for periods

between

j=l

t = T - 1 to t = 1,

(cl

(5)

P,,=Max(D;G)-Q,, where

-I’(.qs,) I d&z”-’ + i PO, i /=I J=l T

C=

d

2 2

I

P,,“-‘ol3wI&J@-’ d

PO,

2 PjJ-‘I(g@,) b#&z’--‘~

at t=O. For probabilistic abandonment Following Bonini [2], recursive this paper as

+q2 i

value the normal probability distribution has been considered. relations for the calculation of the variance of NPV have been derived

P,,[(E+,(s,) I&r>-

w~+GI)

in

I d12

J=l

+ I2q2

+z2 i /=I

s

p,,[(~~+,jS,)IC,)-E(y+,iS,)l~,)!

P,,&(Q

if A,(S,)
(6)

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P.K. De el al. / Capital inoestment ustng multiperiod CAPM

and a,‘(S,)=O, The recursive

relations

ifA,(S,)aP,(S,). are computed

(7) from t = T to t = 0.

backwards

4. Project evaluation with uncertain project life A more realistic situation is considered by taking into account the uncertainty in the life of a project. The probability distribution of the project life can be of various nature. viz., exponential, gamma and normal, and these probability distributions can be approximated (Fig. 1) by triangular distributions [1,3]. Since, pessimistic, modal and optimistic estimates of the random variable completely define a triangular distribution, we believe that it will be relatively easier for the decision maker to specify the triangular distribution. Triangular probability distribution of the project life has been considered in this paper. Let pr be the probability that the project will terminate at t. The method of calculatingp, is given in Appendix B. We have derived the recursive relations to evaluate the expected NPV of a project when the life of the project is uncertain. The relations are: (a) The expression for P,(S,) is the same as eq. (3). (b) P,(s,)

= Max[A,(S,);

&‘(S,);

c/(s,)].

(8)

where

1

/=I

for periods between t = T - 1 to t = 1. (c) The expression for P,, is the same as eq. (5). The variance of NPV, when the life of the project expression: i p,UV( f ) + i p,(UE( 1= I I=1

is uncertain,

can be calculated

from the following

t))2 - (NPVU)‘,

(9)

where UE( t) and UV( t) are the expected value and variance of NPV of the investment respectively when thelifeof theprojectisf(t= 1,2,..., T) years. NPVU is the expected NPV when the life of the project is uncertain. The statistical background for the derivation of expression (9) is given in Appendix C.

EXPONENTIAL

Fig. 1. Various

probability

x

distributions

x

GAMMA

approximated

to triangular

distributions

NORMAL

x

P.K. De et al. / Cupiral investmenr

usrng multlperiod

365

CAPM

Table 2 Input data Year (I) I

2

3

4

Expected cash flow: 1000 1000 1000 1000 Standard deviation of cash flow: 100 120 140 160 Expected abandonment value at the year end: 6200 5700 5180 4580 Standard deviation of abandonment value: 200 190 180 170 Initial investment at [ =O: 6145

Table 3 Computational Correlation coefficient

5

6

7

8

9

10

1000

1000

1000

1000

1000

1000

180

200

220

240

260

280

3980

3300

2570

1780

920

0

160

150

140

130

120

0

results No abandonment

Deterministic abandonment

Probabilistic abandonment

11values

4 values

7 values

0.25

0.35

A. Expected 0.0 0.2 0.5 0.9 1.o

NPV with certain 725 725 725 725 725

project life: 504 504 504 504 504

B. Expected 0.0 0.2 0.5 0.9 1.0

NPV with uncertain -1737 - 1737 -1737 - 1737 -1737

~ -

C. Standard 0.0 0.2 0.5 0.9 1.0

dewation

D. Standard 0.0 0.2 0.5 0.9 1.0

deviation of NPV with uncertain 1383 1329 1397 1342 1429 1374 1531 1473 1577 1519

project 1826 1826 1826 1826 1826

0.25

0.35

0.50

0.25

0.35

0.50

190 190 190 190 190

725 725 729 932 1119

534 535 563 777 955

463 463 463 594 759

750 742 726 858 I045

596 586 554 703 881

471 466 463 531 686

1956 1956 1956 1956 1956

582 582 582 599 619

534 534 534 543 556

463 463 463 465 471

593 588 582 583 601

542 538 534 534 543

469 464 463 463 465

244 271 342 653 752

89 97 188 546 678

89 89 89 442 557

282 300 419 712 813

254 262 285 598 736

264 263 265 456 613

284 297 353 610 711

64 66 163 490 604

69 69 69 342 462

408 410 409 638 759

310 303 277 511 650

260 260 259 369 507

0.50

life:

of NPV with certain 396 379 475 455 639 613 1061 1019 1238 1190

~ ~ ~ ~

project life: 356 428 516 959 1121 project 1252 1264 1296 1391 1435

life

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P. K. De et al. / Caprtal investment usrng mhperiod

CA PM

5. An illustrative example A numerical example is given to illustrate the models developed in Sections 3 and 4. Input data in Table 2 have been taken from Robichek and Van Horne [6], Bonini [2] and De et al. [3]. Other input values required for the computation are, E(R,,,, ,) = 0.12, r = 4%, ui = 0.02, ha,,,,, = 0.02, d = 11, t, = 0. t,,, = 6 and t, = 10. Expected value and variance of NPV of the project have been calculated for different values of 11(0.25,0.35 and 0.50) and correlation coefficient of cash flows, p (0,0.2,0.5,0.9 and 1.0) for both certain and uncertain life of the project. The dynamic programming models are programmed on a Burroughs B6700 system and the results are given in Table 3.

6. Summary and conclusions This paper presents a dynamic programming approach to evaluate an investment considering the abandonment options in a multiperiod capital asset pricing model context. It can be seen from the results of Table2 that the NPV of a project for which the abandonment options are explicitly considered is generally higher than the NPV of the project for which the abandonment options are not taken into account. From the basic valuation formula of Myers and Turnbull ((A-l) in Appendix A), it can be observed that if all other parameters are kept constant, Z is a decreasing function of TJ.Therefore, as n increases, the expected NPV of a project should decrease. This is illustrated in Table 3. It has also been observed that the expected NPV, in case of probabilistic project life, is less than its deterministic counterpart, a result that confirms the usual intuitive feeling. The significance of the present study is its ability to consider uncertainty in the life of a project while taking the abandonment options into account. Inspite of the usual limitations of the multiperiod CAPM approach [5], the methodology presented here provides a framework within which the various uncertainties associated with the evaluation of a project can be effectively considered.

Appendix A Myers and Turnbull [5] have derived the following flows from a project in a multiperiod CAPM context:

expression

to evaluate

the present

worth

of cash

T-l &I=&%

I$")4

(A-1)

2 Z'> /=o

where current

uIM

b

x

expectation

of the cash flow at t = 1, given information

at &;

(1 - hba,,,,,)/(l + r); (1 - X~be,,,,),‘(l + r); life of the project; one period risk free interest rate; is the return on the market portfolio. Market portfolio covariance (i,,iiM ,), where d,,, represents the optimal combination of all risky projects. 6 represents the unanticipated changes in some general economic index; firm-specific constant representing the sensitivity of the disturbance term (bf, + h,) to unanticipated changes in the economic index (f,). fi, is purely a firm or project specific parameter; an exogenous market parameter defined as (E( d,W., , ,) - Y)/u;,, where CJ,$ denotes the over time, thus ensuring a stationary variance of ti,,,,. u$ is assumed to be constant probability distributon for iM.,+ ,;

P.K. De et al. / Cupltul wwestment

usmg multiperwd

: elasticity of expectations representing the proportional the observed changes in (b( + k,) and 0 =G17G 1.

Appendix

367

CAPM

changes

in the expectation

relative

to

B

Let t,. t,,, and t, be the pessimistic, modal and optimistic life of a project. The continuous triangular distribution is converted to a discrete distribution by assuming that P, for t > t, equals to the cumulative probability between (t - 1) and t. This indicates that P, is the area of the segment between (t - 1) and t in the triangular distribution. F;, the cumulative probability at t, is given by

~=(r-t,)2/(t,,-tp)(ta-tp)

fortpeq,,,

(B-1)

F,=

fort,,GtGt,.

(B-2)

1 -(t‘,-r)2/(t,-tt,)(t;,-ttp)

and p, equal to zero when t 4 t,, and t > t r\.

Appendix

C

The basic statistical framework for uncertain project life is described here [ 11. For a general discussion of the statistical concept, the values of q and Z are taken as 1. Let X,, X2,. .,X, are independent and identically distributed random numbers, which represent the cash flows in the present study. T,which represents the life of a project, is a non-negative integer-valued random variable. The mean and variance for independent X’s with random variable T,can be written as E

= i

p, i E(Y), r=I /=I

(C-1)

V

=

5 p, i V(X,)+ /=I ,,= 1

(C-2)

For correlated values of X’s, the expression the variance will be

for the mean will remain

the same, but the expression

for

(C-3) where T’ is the realization respectively.

of

T, and E() and

V( ) are the expected

value

and

variance

operators

References

I11 R.P. Bey, The impact of stochastic project lives on capital budgeting

decisions, in: R.L. Crum and F.G.J. Derkindern, Eds., Cuprtal Budgetmg under Condrtions of CJncertarnt_~(Nijhoff, Boston. 1981) 63-80. I21 C.P. Bonini. Capital investment under uncertainty with abandonment options. .I. Fuuuzcial and Quantltafme Anal. 12 (I) (1977) 39-54. [31 P.K. De, D. Acharya and K.C. Sahu. Estimation of mean and variance of net present value with certain and uncertain project life: a multiperiod CAPM approach, European J. Operatroncrl Rex 8 (4) (1981) 363-368. [41 D.B. Hertz. Risk analysis in capital investment, Harwrd Business Rev. 42 (1) (1964) 95- 106.

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P. K. De et ul. / Capital investment

using multiperiod CA PM

[5] S.C. Myers and SM. Turnbull, Capital budgeting and the capital asset pricing model: good news and bad news, J. Fwumcr _?2(2) (1977) 321-333. [6] A.A. Robichek and J.C. Van Horne, Abandonment value and capital budgeting. J. Fmmce 22 (4) (1967) 577-589. [7] A.A. Robichek and J.C. Van Horne, Abandonment value and capital budgeting: reply. J. Fmznce 24 (1) (1969) 96-97. [8] B. Schwab and P. Lusztig, A note on investment evaluations in light of uncertain future opportunities, J. Fmmce 27 (5) (1972) 1093-l 100. [9] J.C. Van Home, Capital budgeting under conditions of uncertainty as to project life, Engrg. Econom. (1972) 189- 199. [lo] B. Wagle, A statistical analysis of risk in capital investment projects, Operational Res. Quurt. (1967) 13-33.