- Email: [email protected]
On the present-worth moments of serially correlated cash flows
281
Engineering Costs and Produclion Economics, 16 ( 1989 ) 28 l-289
Elsevier Science Publishers B.V., Amsterdam
- Printed in The Netherlands
ON THE PRESENT-WORTH MOMENTS OF SERIALLY CORRELATED CASH FLOWS L.C. Leung Department of General Business Management
and Personnei management, (Hong Kong)
The Chinese University of Hong Kong, Shatin
Y.V. Hui Department
of Statistics, The Chinese University of Hong Kong, Shark (Hong Kong)
and G.A. Fieischer Department
of lndusrrial and Systems Engineering, University of Southern California, Los Angeles, CA (U.S.A.)
ABSTRACT
The net present value (NPV) ofa seriaily correlated cash-flow profile is examined. Given the expert estimates of end-of-period cashljlows, which are assumed to be beta variates, a methodology is developed to determine the~rst four moments of the profiles NPV We first determine the moments of the cash-flows, whereby the NPV expectation results readily. Then a correlative model based on time-series analysis is assumed in order to examine the cash~~o~v dependency: this is required for the determina-
tion qf higher moments. Subsequently, it is shown that the higher moments ofthe NPVcan be determinedfrom the moments ofperiod cashflows and the time-series parameters. Although the betu-distributed ass~rnpt~o~ is used to illustrate the methodology, it is not a requirement in this analysis. Indeed, the approach presented in this paper is applicable to a finite cash-flow prqfile with arbitrarily distributed cash flows, as lorzg us their individual moments can be deteri~ined.
INTRODUCTION
The analysis of a risky investment can be greatly facilitated if its probability density function can be determined or reasonably estimated. But this can be a complex task, made even more difficult when there exists inter-period dependency among cash flows. It is wellknown that if the future cash flows are normal variates, the NPV distribution would also be normal regardless of whether the returns are independent or not. Moreover, if the future cash flows can be assumed to be identically and independently distributed, the NW distribution would also be approximately normal pro-
In the evaluation of investments under risk, a major difficulty lies in the treatment of future stochastic cash flows. In many instances, the expected value of an investment’s net present value (NPV) is commonly used as a measure of the investment’s economic merit. The robustness of the analysis is often enhanced when the objective function incorporates both the expected value and the variance of the NPV, with their relationship described by the decision maker’s utility function. 0167-188X/89/$03.50
0 1989 Elsevier Science Publishers B.V.
282 vided there are a relatively large number of future cash flows. Also, assuming a large number of cash flows, the NPV distribution converges to normality if the cash flows are dependent stationary Markovian random variables. In general, where inter-period dependency exists, other than the aforementioned normal and stationary Markovian cases, the NPV distribution cannot be specified. Detailed discussion of the above results can be found in Hillier [ 1 ] and Bussey [ 2 1. If a cash flow sequence can be described as a function of time with the random parameters being the initial cash flow, its relative timing and its duration, then transform techniques can be employed to arrive at either the closed-form NPV distribution [ 3,4,5 ] or an approximated NPV distribution via its moments [ 6,7]. One feature of such functional form assumption is that the issue of inter-period dependency need not be explicitly addressed; period-to-period dependencies are subsumed in the prescribed functional relationship. Several authors have examined the dependency issue by a time-series approach. Bussey and Stevens [ 81 assumed that the cash-flow profile be described by a first-order autoregressive time-series. They show that once the lag coefficient can be estimated, the mean and variance of the NPV can be computed. Although some difficulty was later suggested regarding Bussey’s simulation-regression procedure which formed the basis for the estimation of the lag coefficient, their work established the time-series methodology as a viable one. Also, Giaccotto [ 91 showed how the NPV’s mean and variance can be computed when the lag coefficient of a first-order autoregressive cashflow series is known. There is a major difference between Giaccotto’s work and that of Bussey and Stevens. Giaccotto’s approach begins with the time-series model. As such, both the mean and the variance of the profile’s NPV are dependent on the parameters of the time-series. Bussey and Stevens begin with the estimates of the period
cash-flows. It is after the mean of the prolile’s NPV is determined from the estimates that the time-series analysis is used to model the dependency structure, which subsequently leads to the evaluation of the variance of the NPV. As a result, the expected NPV from Giaccotto varies with the lag coefficient while that from Bussey and Stevens does not. Although offering no explanation, Giaccotto does point out such a difference in reference to Hillier’s work [ 11, of which Bussey and Stevens is an extension. If it is known a priori that the time-series model fully describes the cash-flow profile, it would appear that Giaccotto’s approach could be justified. However, a crucial question remains: How can the time-series parameters be determined, if not via some estimation procedure on certain available data? If the time-series parameters are indeed the outcome of an estimation procedure, it follows that the expected NPV directly obtained from the data is likely to be more accurate than that obtained subsequently from the model. Bussey and Stevens’ approach is in line with that view. It assumes the existence of available information, derives the NPV mean therefrom, and uses the time-series model only as an approximation to the correlation effect. The Bussey and Stevens approach would appear to be more realistic as it is based on the period-by-period estimates of the cash flows. This paper extends the work of Bussey and Stevens. Given the expert estimates of period cash-flows, which are assumed to be beta variates, a methodology is developed to determine the first four moments of the profile’s NPV. The moments of the period cash-flows are determined, whereby the NPV expectation results readily. In order to derive the NPV’s higher moments, the cash-flow dependency is then modelled using the first-order auto-regressive time-series structure. The parameters of the time series are approximated from expert estimates via the error-in-variable methodology [ lo]. Subsequently, the higher mo-
283 ments of the NPV can be determined from the moments of period cash-flows and the time-series parameters. Although the beta-distributed assumption is believed to offer practical value, it is not a requirement. The methodology presented in this paper is applicable to any finite cash-flow profile with arbitrarily distributed cash flows, as long as their individual moments can be determined. Hence, this approach does not require such restrictive assumptions as normality or a relatively large number of cash Ilows. PROBLEM STATEMENT
Consider a proposed investment which generates a cash flow profile Y, for r=O, 1,2,.X. That is, Y, is a cash flow at the end of period t, with Nperiods in the planning horizon. (In the NPV models which follow, end-of-period cash flows are assumed. This is not strictly necessary, however. If the cash flows are assumed to flow continuously and uniformly throughout the period, for example, the end-of-period equivalent can be determined by multiplying the cash flow by the funds flow conversion factor [i/In ( 1 + i) 1, where i is the effective interest rate for the period. ) In general, no assumption is made as to the degree of inter-period dependency of the cash flows. They may be independent, partially correlated, or perfectly correlated. No general assumption is made as to the probability distributions of the period cash flows. They may be normal or non-normal, continuous or discrete, identically or not identically distributed. It is necessary, however, that the ~u~e~~~ of the distributions of these cash flows be determinable, at least to the extent of the first four moments. In the numerical example which follows, the period cash flows are assumed to be beta distributed. The beta dist~bution is bounded and unimodal, and thus has reasonable similarity to distributions that might be expected under actual conditions. The PERT approximation to
TABLE 1 A serially correlated cash flow profife with beta distributed cash flows* Year
t
Pessimistic value y, (t) x10’
0
1 2 3 4
5 6 7 8 9 10
-30 -32 -8 6 12 12 14 16 17 18 20
Optimistic value t-: (t)
Mean E(Y,)
Variance Var( Y,)
X10”
x106
-22 -26 -3 12 20 20 23 25 26.5 29 30
7.11 4.00 2.78 4.00 7.11 7.11 9.00 9.00 10.31 13.44 II.11
x103
-14 -20 2 18 28 2s 32 34 36 40 40
*This example is based on one appearing in [ 2, p. 3861. In the original version, the means and variances at the yeariy cash flows are computed from optimistic, pessimistic and most likely estimates using the PERT methodology. However, in this example, all values in the table, including the means and variances, are assumed to have been estimated.
the beta distribution, widely discussed in the literature, cf., Bussey [ 21, Hillier [ 11 and Wagale [ 111, requires only that the most optimistic, most pessimistic, and the most likely values be estimated, and from which the mean and variance of the distribution are approximated. This approximation scheme can produce a significant error, however, and it is not used here. Instead, it is assumed that expert estimates are available for the pessimistic and optimistic values of the period cash flows, Y,( t ) and Y,(t), as well as the mean, ,LL*, and the variance, af, of the beta distributed cash flows in period t. A numerical example, based on one appearing in [ 21 f is given in Table 1 for a lo-year planning horizon. The objective is to derive the first four moments of the distribution of the NPV arising from this sequence, or profile, of cash flows. The four moments will then be used to approximate the NPV distribution. The NPV of the cash-flow profile is given by
284 where al= ( 1 +i) -t, the discount factor, and i is the risk-free discounting rate. The mth order moment of Z is
first moment (m = 1 ), the mean of Z, can be found directly from the available data, specifically, ,u, =E[ Y, 1. Returning to the example in Table 1, and assuming i=O. 15, that is, at= (1.15)-C E(Z)=
Equation (2) can be expanded to become the summation over all combinations of terms a/oO . ..&E( Y$ . . . Y$) where j,,, j, ,..., j, are nonnegative integers taken from 1 to m such that j, +j, + . . . +j,=m. As such, the first four moments can be expressed as: m= 1, E(Z)=
;a,E(Y,)
p=o
(3a)
m=2, E(Z’)=
(4)
For the determination of the second, third, and fourth moments, both the moments of Y,, i.e., E( Y<) and the expected value of the crossproduct
terms, E ( t!. Y; ) , are required.
; a;E(Y;) p=o
will first examine E( Y{). MOMENTS
OF Y;S
Y,=(O,-P,)W,$P,
(5)
where IV, is beta distributed over the interval [ 0,l ] with parameters cy and p, 0, is the optimistic value and P, is the pessimistic value. The Appendix shows the details on the derivation of (Y and p. To determine E( Y<), j= 1,2,3,4, from eqn. (5):
m=3, = 5 a;E( Y;)
E( Y,)= (0, -J’t)E( w,) +K (3c)
E( Y:) = (0, -P,)‘E(
(6a)
W:)
+2(0,-P,)P,E(
and m=4,
W,)+P:
(6b)
E(Y:)=(O,-p03E(W:)
E(Z4) = f a$?( r;, p=o
+w:E(
We
First the parameters, LYand /3, of the underlying beta distribution for Y,‘s are determined. Here, Y, may be rewritten as: (3b)
E(Z))
&z,p,=26,056 *=o
+3(0,-P,)2P,E(
W:)
+3(0,-P,)P:E(
W,)+P:
Y, Yi) 1 E( Y;‘) = (0, -Pr)4E(
(6~)
W:)
+4(0,-P,)3P,E(
Iv:)
+6(0,-P,)‘P:E(
W:)
+4(0,-P,)P:E(
W,)+P;
+u,a$2,E( Y, r; Y,)
++,U(Y;Y,Y,)l +24
CcCC uauaE(YYYY) PYU” PIUU O
It is well-known
that (see, for example
(6d)
[ 12 ] ) :
(3d)
(7a) As may be noted from eqns. ( 3a-d),
only the
285 TABLE 2 First four moments for the period cash flows Year
t 0
1 2 3 4 5 6 I 8 9 10
E( Y,) x103
E(Y:) x109
E( Y:, x lOI
E(K’) x 1o’8
- 22.0 - 26.0 -3.0 12.0 20.0 20.0 23.0 25.0 26.5 29.0 30.0
0.49111 0.68000 0.01178 0.14800 0.407 11 0.407 11 0.53800 0.63400 0.71256 0.85444 0.91111
-11.117 - 17.888 -0.052 1.872 8.427 8.427 12.788 16.300 19.429 25.558 28.000
0.25503 0.47324 0.00025 0.02423 0.17719 0.17719 0.3086 1 0.42457 0.53686 0.77554 0.87030
cash-flow profile, 1191< 1, e, is the error at period t, with E( e,) =O and Var [ e,] = c?, and is symmetrically distributed, and 6 is the trend of the cash-flow profile, a constant. With this assumption, it is now possible to N
(9)
Now, define gs(n) = r,,,,
(7b)
(7c)
(7d) Using eqns. (6a-d) and (7a-d), we compute E( Y<) forj= 1,2,3,4. The results are shown in Table 2 for the numerical example. EXPECTED VALUE OF THE CROSSPRODUCT TERMS To evaluate product
terms,
the expected E ( t!OY’:),
+e,+6,
where 6’is the one-period
it is necessary
to
(8)
lag coefficient
(10)
Substitute all the Y’s in eqn. ( 10) with the firstorder autoregressive equations in eqns. (8 ); also replacing the Y’s with g’s recursively, we have
g,(n)=eg,(n-l)+6+e,+. Equation E
(11)
(9) can now be expressed =E{ [g,(O) l”M
1) l“+'...ks(l-s)
as I"}
I-S
value of the cross-
t=O,...,N
n=O,...,l-s
=E
assess the dependency structure of Y,‘s. Here, a correlative model is assumed for the cashflow profile. Specifically, it is assumed that the cash-flow profile { YO, Y, ,..., YN} has an underlying autocorrelated structure of a first order autoregressive time-series. (See also Montgomery and Johnson [ 13 ] for time-series models. ) That is, Y,=W_,
)
E ( n Y$ . We let j, be the smallest t=o non zero jr, and j, be the largest non-zero jI, t=O ,..., N. That is, evaluate
for the
n k>(t) I,+> ,=s
(12)
Equation ( 12) is implicitly in terms of 0, 6; E( Y,), ..., E( Y,“); and E(e),..., E(e”-I)). Next eqns. ( 11) and ( 12) are used to derive either the closed-form expressions or the recursive relationships for all the terms which appear in the second, third, and fourth present-worth moment expressions, i.e., eqns. (3b), (3c), and (3d) respectively. Both the closed-form expressions and initial conditions of the recursive relationships will only involve the following terms: 0, 6, E( e2), and E ( Y’,). Although E(e3) is needed to compute the fourth moment, it has a zero value, resulting from the assumption that the error term is symmetrically distributed.
286 I?@*): From eqn. (3b), it is recognized that E( 2* ) consists of terms which can be categorized as follows: E( Y,“), and E( YSYy,),S-CL The evaluation of these terms are, (i)E(Y~),s=1,2,...,N.Thesevalueshavebeen determined previously. (ii)E(Y,YI),s
=OE{ Y&l-s-
(iii) E( Y,Y:), s
and
Y,)
l)}+a(
+{P+E(ef)jE(Y,) +286E( Y,Y,_,)
=@2E(Y&(f-s-2))
E( Y, Y:) can be computed recursively from eqns. (13) and (14a). (iv) E( Y,Y,Y,), s
+ME( Y,) +6E( Y,) =P%y
Yf) +&lq Y,)
t 14b)
I-s- I r, 8’ J-0
(13)
E
Thus, E( Y, Y,Y,) can be computed from eqn. (14b). E(z”l): From eqn. (3d ), it is recognized that E(Z4) consists of terms which can be categorized as follows: E(Y!); E(YzY,), s
=eEIY:g,(l-f-l)j+S14’(Y:) (ii) E( Y,3Y,) =B’-‘E(
=&E{Y:[Sg,(l-s-2)+6+e,_,]) i”SE( Y,2)
Ys”)
+6E( y:,$y
s
tl5a)
=f?%{Y,2g,(l-s-2)) +f%E( Y,z)+Z(
(iir) E( Y,Yj)=Q3E(
Y,“) l-z-i
=O’-sE{Y:j+&F(
Ys”) c
@’
YsYLt)
+ 38%E( r, r:_ , )
j-0
+36E(Y,YI_,){S2+E(e:)J +E( Y,){S’+36E(ef)]
sci
(15b)
287 (iv)E(YjY:)=e2E(Y~Y~_,)+E(Yj) is’+-wd)i +2e&!?( Y:Y,_,)
s-cl
(v) E( YsY,Yt,)=fZ12E( Y,Y,Y;_,)
r,=x,+z,
+wYsY,){62we2,)} $28&E( Y,Y,Y,_,)
s-cl
(vi)E(Y,YfY,)=e”-‘E(Y,Y:) +sE( Y,Yf) 1-p-l
1-e (vii) E( YSY,Y,) = e%q
S
Y%Y:)
+dE(Y:YI) 1-eu-’ i-e
cedure is a substantial research topic in its own right, the details are omitted here; only the conceptual framework is outlined concerning how this methodology can be applied to the present situation. Let
S
(16)
where X,=E( Y,) and Z, is the error from the mean value X,. Note that the Z,‘s are independent and that E(Z’;), j= 1,2 ,..., m and t=O,l,...,Ncan be derived from E( Y{). From eqns. (8) and ( 16), the following expression can be established: x,-ex,_,
-6=e,-z,+ezlp,
(17)
We can now estimate 8 and 6 using the errorin-variable methodology. Once the estimates of 8 and 6 are found: we can proceed to estimate E(e;) via the methodology of moments estimation [ 12, p. 2531. Again, considering eqn. (17), we write
(viii)E(YsY,YuYh)=eh-‘E(Y,Y,Y:) >
+6E(YsY,Yu) 1
~
_@-u i-e
=tf,E(X,-exl_,
= -8)’
s
All the terms which appear in the second, third, and fourth present-worth moment expression are now expressed either explicitly or implicitly in terms of 19, 6, E(e2), and E( Y’,), j = 1,2,3,4. The E( Y/;)‘s have previously been determined. In the next section, we will discuss the framework of an approach based on which the time-series parameters 0, 6, and E(e*) can be approximated from the expert estimates. ESTIMATION OF THE TIME-SERIES PARAMETERS
The estimation procedure for the parameters in the time-series model, 0, S, and E( e* ), is based on the error-in-variable approach proposed in [ lo]. Since such an estimation pro-
~iqe,-z,+ezl_,y
(18)
1=1
N
Using C (X,-0X,_,
-S)‘,j=
1,2 ,..., m, wecan
I=1
estimate E(d),
j= 1,2,..., m successively.
RESULTS
Based on the data provided in Table 1, the estimation procedure described above gives the following time-series parameters for our numerical example: 19=0.4, 6=0.184x 105, and E(e2)=0.13989x109. We are now ready to evaluate E( Z’), j= 2,3,4. Using eqns. (3b-d), (13), (14a-c), and (15a-g), as well as the now-known values of E( Y{), 19,6, E( e’), the second, third, and fourth moments of the profile’s NPV are:
288 E(Z2)=3.5773x
lo9
REFERENCES 1
E(Z4)=202.27x
10”
Recall, in eqn. (4) we found E(Z) =24,056. With knowledge of the first four moments, it is now possible to approximate the present-worth distribution via the Pearson methodology of distribution fitting [ 141. The Pearson procedure only requires the first four moments. With knowledge of the distribution, the analyst is then in position to make probability (risk) statements concerning the NPV random variable. CONCLUSIONS
In the analysis of a risky investment, interperiod dependency within the cash-flow profile causes considerable difficulty in the determination of the investiment’s NPV. The proposed procedure begins with the expert estimates of the profile’s period cash-flows. These cash flows may follow any distribution as long as the moments of the individual cash flows can be estimated. (In this paper, we have assumed beta distributed random variables. ) Upon obtaining the expectation of the NPV directly from the estimated cash flows, a firstorder autoregressive time-series model is assumed to approximate the profile’s dependency structure; this is necessary in order to determine the higher moments of the NPV. The NPV’s higher moments can be determined from the moments of the period cash-flows and the derived time-series parameters. The period cash flows are assumed to follow the beta distribution, in part because of the attractive characteristics of this distribution: boundedness and single mode. The beta assumption is not essential, however. The methodology is applicable to arbitrary period cash flow distributions, as long as their moments can be estimated.
Hillier, F.S., 1969. The Evaluation of Risky Interrelated Investments. North-Holland, Amsterdam. 2 Bussey, L.E., 1978 The Economic Analysis of Industrial Projects. Prentice-Hall, Englewood Cliffs, NJ. 3 Barnes, J.W., Zinn, C.D. and Eldred, B.S., 1978. A methodology for obtaining the probability density function of the present worth of probabilistic cash flow protiles. AIIE Trans., 10 (3): 226-236. 4 Young, D. and Contrerars, L.E., 1975. Expected present worth of cash flows under uncertain timing. Eng. Econ., 20 (4): 257-268. 5 Zinn, C.D. and Lesso, W.G., 1977. A probabilistic approach to risk analysis in capital investment projects. Eng. Econ. 22 (4): 239-260. 6 Tanchoco, J.M.A. and Buck, J.R., 197 1. A closed-form methodology for computing present worth statistics of risky discrete cash-flows. AIIE Trans. 9 (3): 278-287. 7 Tanchoco, J.M.A., Buck, J.R. and Leung, L.C., 1981. Modeling and discounting of continuous cash flows under risk. Eng. Costs Prod. Econ., 5: 205-215. 8 Bussey, L.E. and Stevens, Jr., G.T., 1972. Formulating correlated cash flow streams. Eng. Econ., 18 ( 1): 1-3 1. 9 Giaccotto, C., 1984. A simplified approach to risk analysis in capital budgeting with serially correlated cash flows. Econ., 29 (4): 273-286. 10 Hannan, E.J., 1963. Regression for time series with errors of measurements. Biometrika, 50 (3): 292-302. 11 Wagle, B., 1967. A statistical analysis of risks in capital investment projects. Oper. Res. Q., 18 ( 1): 13-33. 12 Hogg, R.V. and Craig, A.T., 1970. Introduction of Mathematical Statistics, 3rd edn. Macmillan, NJ. 13 Montgomery, D.C. and Johnson, L.A., 1976. Forecasting and Time Series Analysis. McGraw-Hill, NY. 14 Pearson, K., 1942. Karl Pearson’s Early Statistical Papers, Cambridge University Press, Oxford. Referenced by J.A. Greenwood and H.O. Hartley, 1962. Guide to Tables in Mathematical Statistics. Princeton University Press, Princeton, NJ. (Received
June 28,
APPENDIX
1988;
accepted
August
4,
1988)
1
From the underlying beta distribution, have p=(U-L)L-+L a+B
we
(A.1)
and a/? ~2=(U-L)2~ru+p+l)(a+p)2
(A.21
289 where L = Pt,U= O,,,LL= mean, g2 = variance, and cy and p are parameters of the beta distriand Let bution. r= (P---)l(U-L) Q= ( U-L)'/a2.Then, eqns. (A. 1) and (A.2) can be written as
LY=Q(l-17)I~-(l+(1-)1)/)1)2 C1+(1-V)/?13
(A.3)
P=a(l-rl)lrl
(A.4)
Engineering Costs and Produclion Economics, 16 ( 1989 ) 28 l-289
Elsevier Science Publishers B.V., Amsterdam
- Printed in The Netherlands
ON THE PRESENT-WORTH MOMENTS OF SERIALLY CORRELATED CASH FLOWS L.C. Leung Department of General Business Management
and Personnei management, (Hong Kong)
The Chinese University of Hong Kong, Shatin
Y.V. Hui Department
of Statistics, The Chinese University of Hong Kong, Shark (Hong Kong)
and G.A. Fieischer Department
of lndusrrial and Systems Engineering, University of Southern California, Los Angeles, CA (U.S.A.)
ABSTRACT
The net present value (NPV) ofa seriaily correlated cash-flow profile is examined. Given the expert estimates of end-of-period cashljlows, which are assumed to be beta variates, a methodology is developed to determine the~rst four moments of the profiles NPV We first determine the moments of the cash-flows, whereby the NPV expectation results readily. Then a correlative model based on time-series analysis is assumed in order to examine the cash~~o~v dependency: this is required for the determina-
tion qf higher moments. Subsequently, it is shown that the higher moments ofthe NPVcan be determinedfrom the moments ofperiod cashflows and the time-series parameters. Although the betu-distributed ass~rnpt~o~ is used to illustrate the methodology, it is not a requirement in this analysis. Indeed, the approach presented in this paper is applicable to a finite cash-flow prqfile with arbitrarily distributed cash flows, as lorzg us their individual moments can be deteri~ined.
INTRODUCTION
The analysis of a risky investment can be greatly facilitated if its probability density function can be determined or reasonably estimated. But this can be a complex task, made even more difficult when there exists inter-period dependency among cash flows. It is wellknown that if the future cash flows are normal variates, the NPV distribution would also be normal regardless of whether the returns are independent or not. Moreover, if the future cash flows can be assumed to be identically and independently distributed, the NW distribution would also be approximately normal pro-
In the evaluation of investments under risk, a major difficulty lies in the treatment of future stochastic cash flows. In many instances, the expected value of an investment’s net present value (NPV) is commonly used as a measure of the investment’s economic merit. The robustness of the analysis is often enhanced when the objective function incorporates both the expected value and the variance of the NPV, with their relationship described by the decision maker’s utility function. 0167-188X/89/$03.50
0 1989 Elsevier Science Publishers B.V.
282 vided there are a relatively large number of future cash flows. Also, assuming a large number of cash flows, the NPV distribution converges to normality if the cash flows are dependent stationary Markovian random variables. In general, where inter-period dependency exists, other than the aforementioned normal and stationary Markovian cases, the NPV distribution cannot be specified. Detailed discussion of the above results can be found in Hillier [ 1 ] and Bussey [ 2 1. If a cash flow sequence can be described as a function of time with the random parameters being the initial cash flow, its relative timing and its duration, then transform techniques can be employed to arrive at either the closed-form NPV distribution [ 3,4,5 ] or an approximated NPV distribution via its moments [ 6,7]. One feature of such functional form assumption is that the issue of inter-period dependency need not be explicitly addressed; period-to-period dependencies are subsumed in the prescribed functional relationship. Several authors have examined the dependency issue by a time-series approach. Bussey and Stevens [ 81 assumed that the cash-flow profile be described by a first-order autoregressive time-series. They show that once the lag coefficient can be estimated, the mean and variance of the NPV can be computed. Although some difficulty was later suggested regarding Bussey’s simulation-regression procedure which formed the basis for the estimation of the lag coefficient, their work established the time-series methodology as a viable one. Also, Giaccotto [ 91 showed how the NPV’s mean and variance can be computed when the lag coefficient of a first-order autoregressive cashflow series is known. There is a major difference between Giaccotto’s work and that of Bussey and Stevens. Giaccotto’s approach begins with the time-series model. As such, both the mean and the variance of the profile’s NPV are dependent on the parameters of the time-series. Bussey and Stevens begin with the estimates of the period
cash-flows. It is after the mean of the prolile’s NPV is determined from the estimates that the time-series analysis is used to model the dependency structure, which subsequently leads to the evaluation of the variance of the NPV. As a result, the expected NPV from Giaccotto varies with the lag coefficient while that from Bussey and Stevens does not. Although offering no explanation, Giaccotto does point out such a difference in reference to Hillier’s work [ 11, of which Bussey and Stevens is an extension. If it is known a priori that the time-series model fully describes the cash-flow profile, it would appear that Giaccotto’s approach could be justified. However, a crucial question remains: How can the time-series parameters be determined, if not via some estimation procedure on certain available data? If the time-series parameters are indeed the outcome of an estimation procedure, it follows that the expected NPV directly obtained from the data is likely to be more accurate than that obtained subsequently from the model. Bussey and Stevens’ approach is in line with that view. It assumes the existence of available information, derives the NPV mean therefrom, and uses the time-series model only as an approximation to the correlation effect. The Bussey and Stevens approach would appear to be more realistic as it is based on the period-by-period estimates of the cash flows. This paper extends the work of Bussey and Stevens. Given the expert estimates of period cash-flows, which are assumed to be beta variates, a methodology is developed to determine the first four moments of the profile’s NPV. The moments of the period cash-flows are determined, whereby the NPV expectation results readily. In order to derive the NPV’s higher moments, the cash-flow dependency is then modelled using the first-order auto-regressive time-series structure. The parameters of the time series are approximated from expert estimates via the error-in-variable methodology [ lo]. Subsequently, the higher mo-
283 ments of the NPV can be determined from the moments of period cash-flows and the time-series parameters. Although the beta-distributed assumption is believed to offer practical value, it is not a requirement. The methodology presented in this paper is applicable to any finite cash-flow profile with arbitrarily distributed cash flows, as long as their individual moments can be determined. Hence, this approach does not require such restrictive assumptions as normality or a relatively large number of cash Ilows. PROBLEM STATEMENT
Consider a proposed investment which generates a cash flow profile Y, for r=O, 1,2,.X. That is, Y, is a cash flow at the end of period t, with Nperiods in the planning horizon. (In the NPV models which follow, end-of-period cash flows are assumed. This is not strictly necessary, however. If the cash flows are assumed to flow continuously and uniformly throughout the period, for example, the end-of-period equivalent can be determined by multiplying the cash flow by the funds flow conversion factor [i/In ( 1 + i) 1, where i is the effective interest rate for the period. ) In general, no assumption is made as to the degree of inter-period dependency of the cash flows. They may be independent, partially correlated, or perfectly correlated. No general assumption is made as to the probability distributions of the period cash flows. They may be normal or non-normal, continuous or discrete, identically or not identically distributed. It is necessary, however, that the ~u~e~~~ of the distributions of these cash flows be determinable, at least to the extent of the first four moments. In the numerical example which follows, the period cash flows are assumed to be beta distributed. The beta dist~bution is bounded and unimodal, and thus has reasonable similarity to distributions that might be expected under actual conditions. The PERT approximation to
TABLE 1 A serially correlated cash flow profife with beta distributed cash flows* Year
t
Pessimistic value y, (t) x10’
0
1 2 3 4
5 6 7 8 9 10
-30 -32 -8 6 12 12 14 16 17 18 20
Optimistic value t-: (t)
Mean E(Y,)
Variance Var( Y,)
X10”
x106
-22 -26 -3 12 20 20 23 25 26.5 29 30
7.11 4.00 2.78 4.00 7.11 7.11 9.00 9.00 10.31 13.44 II.11
x103
-14 -20 2 18 28 2s 32 34 36 40 40
*This example is based on one appearing in [ 2, p. 3861. In the original version, the means and variances at the yeariy cash flows are computed from optimistic, pessimistic and most likely estimates using the PERT methodology. However, in this example, all values in the table, including the means and variances, are assumed to have been estimated.
the beta distribution, widely discussed in the literature, cf., Bussey [ 21, Hillier [ 11 and Wagale [ 111, requires only that the most optimistic, most pessimistic, and the most likely values be estimated, and from which the mean and variance of the distribution are approximated. This approximation scheme can produce a significant error, however, and it is not used here. Instead, it is assumed that expert estimates are available for the pessimistic and optimistic values of the period cash flows, Y,( t ) and Y,(t), as well as the mean, ,LL*, and the variance, af, of the beta distributed cash flows in period t. A numerical example, based on one appearing in [ 21 f is given in Table 1 for a lo-year planning horizon. The objective is to derive the first four moments of the distribution of the NPV arising from this sequence, or profile, of cash flows. The four moments will then be used to approximate the NPV distribution. The NPV of the cash-flow profile is given by
284 where al= ( 1 +i) -t, the discount factor, and i is the risk-free discounting rate. The mth order moment of Z is
first moment (m = 1 ), the mean of Z, can be found directly from the available data, specifically, ,u, =E[ Y, 1. Returning to the example in Table 1, and assuming i=O. 15, that is, at= (1.15)-C E(Z)=
Equation (2) can be expanded to become the summation over all combinations of terms a/oO . ..&E( Y$ . . . Y$) where j,,, j, ,..., j, are nonnegative integers taken from 1 to m such that j, +j, + . . . +j,=m. As such, the first four moments can be expressed as: m= 1, E(Z)=
;a,E(Y,)
p=o
(3a)
m=2, E(Z’)=
(4)
For the determination of the second, third, and fourth moments, both the moments of Y,, i.e., E( Y<) and the expected value of the crossproduct
terms, E ( t!. Y; ) , are required.
; a;E(Y;) p=o
will first examine E( Y{). MOMENTS
OF Y;S
Y,=(O,-P,)W,$P,
(5)
where IV, is beta distributed over the interval [ 0,l ] with parameters cy and p, 0, is the optimistic value and P, is the pessimistic value. The Appendix shows the details on the derivation of (Y and p. To determine E( Y<), j= 1,2,3,4, from eqn. (5):
m=3, = 5 a;E( Y;)
E( Y,)= (0, -J’t)E( w,) +K (3c)
E( Y:) = (0, -P,)‘E(
(6a)
W:)
+2(0,-P,)P,E(
and m=4,
W,)+P:
(6b)
E(Y:)=(O,-p03E(W:)
E(Z4) = f a$?( r;, p=o
+w:E(
We
First the parameters, LYand /3, of the underlying beta distribution for Y,‘s are determined. Here, Y, may be rewritten as: (3b)
E(Z))
&z,p,=26,056 *=o
+3(0,-P,)2P,E(
W:)
+3(0,-P,)P:E(
W,)+P:
Y, Yi) 1 E( Y;‘) = (0, -Pr)4E(
(6~)
W:)
+4(0,-P,)3P,E(
Iv:)
+6(0,-P,)‘P:E(
W:)
+4(0,-P,)P:E(
W,)+P;
+u,a$2,E( Y, r; Y,)
++,U(Y;Y,Y,)l +24
CcCC uauaE(YYYY) PYU” PIUU O
It is well-known
that (see, for example
(6d)
[ 12 ] ) :
(3d)
(7a) As may be noted from eqns. ( 3a-d),
only the
285 TABLE 2 First four moments for the period cash flows Year
t 0
1 2 3 4 5 6 I 8 9 10
E( Y,) x103
E(Y:) x109
E( Y:, x lOI
E(K’) x 1o’8
- 22.0 - 26.0 -3.0 12.0 20.0 20.0 23.0 25.0 26.5 29.0 30.0
0.49111 0.68000 0.01178 0.14800 0.407 11 0.407 11 0.53800 0.63400 0.71256 0.85444 0.91111
-11.117 - 17.888 -0.052 1.872 8.427 8.427 12.788 16.300 19.429 25.558 28.000
0.25503 0.47324 0.00025 0.02423 0.17719 0.17719 0.3086 1 0.42457 0.53686 0.77554 0.87030
cash-flow profile, 1191< 1, e, is the error at period t, with E( e,) =O and Var [ e,] = c?, and is symmetrically distributed, and 6 is the trend of the cash-flow profile, a constant. With this assumption, it is now possible to N
(9)
Now, define gs(n) = r,,,,
(7b)
(7c)
(7d) Using eqns. (6a-d) and (7a-d), we compute E( Y<) forj= 1,2,3,4. The results are shown in Table 2 for the numerical example. EXPECTED VALUE OF THE CROSSPRODUCT TERMS To evaluate product
terms,
the expected E ( t!OY’:),
+e,+6,
where 6’is the one-period
it is necessary
to
(8)
lag coefficient
(10)
Substitute all the Y’s in eqn. ( 10) with the firstorder autoregressive equations in eqns. (8 ); also replacing the Y’s with g’s recursively, we have
g,(n)=eg,(n-l)+6+e,+. Equation E
(11)
(9) can now be expressed =E{ [g,(O) l”M
1) l“+'...ks(l-s)
as I"}
I-S
value of the cross-
t=O,...,N
n=O,...,l-s
=E
assess the dependency structure of Y,‘s. Here, a correlative model is assumed for the cashflow profile. Specifically, it is assumed that the cash-flow profile { YO, Y, ,..., YN} has an underlying autocorrelated structure of a first order autoregressive time-series. (See also Montgomery and Johnson [ 13 ] for time-series models. ) That is, Y,=W_,
)
E ( n Y$ . We let j, be the smallest t=o non zero jr, and j, be the largest non-zero jI, t=O ,..., N. That is, evaluate
for the
n k>(t) I,+> ,=s
(12)
Equation ( 12) is implicitly in terms of 0, 6; E( Y,), ..., E( Y,“); and E(e),..., E(e”-I)). Next eqns. ( 11) and ( 12) are used to derive either the closed-form expressions or the recursive relationships for all the terms which appear in the second, third, and fourth present-worth moment expressions, i.e., eqns. (3b), (3c), and (3d) respectively. Both the closed-form expressions and initial conditions of the recursive relationships will only involve the following terms: 0, 6, E( e2), and E ( Y’,). Although E(e3) is needed to compute the fourth moment, it has a zero value, resulting from the assumption that the error term is symmetrically distributed.
286 I?@*): From eqn. (3b), it is recognized that E( 2* ) consists of terms which can be categorized as follows: E( Y,“), and E( YSYy,),S-CL The evaluation of these terms are, (i)E(Y~),s=1,2,...,N.Thesevalueshavebeen determined previously. (ii)E(Y,YI),s
=OE{ Y&l-s-
(iii) E( Y,Y:), s
and
Y,)
l)}+a(
+{P+E(ef)jE(Y,) +286E( Y,Y,_,)
=@2E(Y&(f-s-2))
E( Y, Y:) can be computed recursively from eqns. (13) and (14a). (iv) E( Y,Y,Y,), s
+ME( Y,) +6E( Y,) =P%y
Yf) +&lq Y,)
t 14b)
I-s- I r, 8’ J-0
(13)
E
Thus, E( Y, Y,Y,) can be computed from eqn. (14b). E(z”l): From eqn. (3d ), it is recognized that E(Z4) consists of terms which can be categorized as follows: E(Y!); E(YzY,), s
=eEIY:g,(l-f-l)j+S14’(Y:) (ii) E( Y,3Y,) =B’-‘E(
=&E{Y:[Sg,(l-s-2)+6+e,_,]) i”SE( Y,2)
Ys”)
+6E( y:,$y
s
tl5a)
=f?%{Y,2g,(l-s-2)) +f%E( Y,z)+Z(
(iir) E( Y,Yj)=Q3E(
Y,“) l-z-i
=O’-sE{Y:j+&F(
Ys”) c
@’
YsYLt)
+ 38%E( r, r:_ , )
j-0
+36E(Y,YI_,){S2+E(e:)J +E( Y,){S’+36E(ef)]
sci
(15b)
287 (iv)E(YjY:)=e2E(Y~Y~_,)+E(Yj) is’+-wd)i +2e&!?( Y:Y,_,)
s-cl
(v) E( YsY,Yt,)=fZ12E( Y,Y,Y;_,)
r,=x,+z,
+wYsY,){62we2,)} $28&E( Y,Y,Y,_,)
s-cl
(vi)E(Y,YfY,)=e”-‘E(Y,Y:) +sE( Y,Yf) 1-p-l
1-e (vii) E( YSY,Y,) = e%q
S
Y%Y:)
+dE(Y:YI) 1-eu-’ i-e
cedure is a substantial research topic in its own right, the details are omitted here; only the conceptual framework is outlined concerning how this methodology can be applied to the present situation. Let
S
(16)
where X,=E( Y,) and Z, is the error from the mean value X,. Note that the Z,‘s are independent and that E(Z’;), j= 1,2 ,..., m and t=O,l,...,Ncan be derived from E( Y{). From eqns. (8) and ( 16), the following expression can be established: x,-ex,_,
-6=e,-z,+ezlp,
(17)
We can now estimate 8 and 6 using the errorin-variable methodology. Once the estimates of 8 and 6 are found: we can proceed to estimate E(e;) via the methodology of moments estimation [ 12, p. 2531. Again, considering eqn. (17), we write
(viii)E(YsY,YuYh)=eh-‘E(Y,Y,Y:) >
+6E(YsY,Yu) 1
~
_@-u i-e
=tf,E(X,-exl_,
= -8)’
s
All the terms which appear in the second, third, and fourth present-worth moment expression are now expressed either explicitly or implicitly in terms of 19, 6, E(e2), and E( Y’,), j = 1,2,3,4. The E( Y/;)‘s have previously been determined. In the next section, we will discuss the framework of an approach based on which the time-series parameters 0, 6, and E(e*) can be approximated from the expert estimates. ESTIMATION OF THE TIME-SERIES PARAMETERS
The estimation procedure for the parameters in the time-series model, 0, S, and E( e* ), is based on the error-in-variable approach proposed in [ lo]. Since such an estimation pro-
~iqe,-z,+ezl_,y
(18)
1=1
N
Using C (X,-0X,_,
-S)‘,j=
1,2 ,..., m, wecan
I=1
estimate E(d),
j= 1,2,..., m successively.
RESULTS
Based on the data provided in Table 1, the estimation procedure described above gives the following time-series parameters for our numerical example: 19=0.4, 6=0.184x 105, and E(e2)=0.13989x109. We are now ready to evaluate E( Z’), j= 2,3,4. Using eqns. (3b-d), (13), (14a-c), and (15a-g), as well as the now-known values of E( Y{), 19,6, E( e’), the second, third, and fourth moments of the profile’s NPV are:
288 E(Z2)=3.5773x
lo9
REFERENCES 1
E(Z4)=202.27x
10”
Recall, in eqn. (4) we found E(Z) =24,056. With knowledge of the first four moments, it is now possible to approximate the present-worth distribution via the Pearson methodology of distribution fitting [ 141. The Pearson procedure only requires the first four moments. With knowledge of the distribution, the analyst is then in position to make probability (risk) statements concerning the NPV random variable. CONCLUSIONS
In the analysis of a risky investment, interperiod dependency within the cash-flow profile causes considerable difficulty in the determination of the investiment’s NPV. The proposed procedure begins with the expert estimates of the profile’s period cash-flows. These cash flows may follow any distribution as long as the moments of the individual cash flows can be estimated. (In this paper, we have assumed beta distributed random variables. ) Upon obtaining the expectation of the NPV directly from the estimated cash flows, a firstorder autoregressive time-series model is assumed to approximate the profile’s dependency structure; this is necessary in order to determine the higher moments of the NPV. The NPV’s higher moments can be determined from the moments of the period cash-flows and the derived time-series parameters. The period cash flows are assumed to follow the beta distribution, in part because of the attractive characteristics of this distribution: boundedness and single mode. The beta assumption is not essential, however. The methodology is applicable to arbitrary period cash flow distributions, as long as their moments can be estimated.
Hillier, F.S., 1969. The Evaluation of Risky Interrelated Investments. North-Holland, Amsterdam. 2 Bussey, L.E., 1978 The Economic Analysis of Industrial Projects. Prentice-Hall, Englewood Cliffs, NJ. 3 Barnes, J.W., Zinn, C.D. and Eldred, B.S., 1978. A methodology for obtaining the probability density function of the present worth of probabilistic cash flow protiles. AIIE Trans., 10 (3): 226-236. 4 Young, D. and Contrerars, L.E., 1975. Expected present worth of cash flows under uncertain timing. Eng. Econ., 20 (4): 257-268. 5 Zinn, C.D. and Lesso, W.G., 1977. A probabilistic approach to risk analysis in capital investment projects. Eng. Econ. 22 (4): 239-260. 6 Tanchoco, J.M.A. and Buck, J.R., 197 1. A closed-form methodology for computing present worth statistics of risky discrete cash-flows. AIIE Trans. 9 (3): 278-287. 7 Tanchoco, J.M.A., Buck, J.R. and Leung, L.C., 1981. Modeling and discounting of continuous cash flows under risk. Eng. Costs Prod. Econ., 5: 205-215. 8 Bussey, L.E. and Stevens, Jr., G.T., 1972. Formulating correlated cash flow streams. Eng. Econ., 18 ( 1): 1-3 1. 9 Giaccotto, C., 1984. A simplified approach to risk analysis in capital budgeting with serially correlated cash flows. Econ., 29 (4): 273-286. 10 Hannan, E.J., 1963. Regression for time series with errors of measurements. Biometrika, 50 (3): 292-302. 11 Wagle, B., 1967. A statistical analysis of risks in capital investment projects. Oper. Res. Q., 18 ( 1): 13-33. 12 Hogg, R.V. and Craig, A.T., 1970. Introduction of Mathematical Statistics, 3rd edn. Macmillan, NJ. 13 Montgomery, D.C. and Johnson, L.A., 1976. Forecasting and Time Series Analysis. McGraw-Hill, NY. 14 Pearson, K., 1942. Karl Pearson’s Early Statistical Papers, Cambridge University Press, Oxford. Referenced by J.A. Greenwood and H.O. Hartley, 1962. Guide to Tables in Mathematical Statistics. Princeton University Press, Princeton, NJ. (Received
June 28,
APPENDIX
1988;
accepted
August
4,
1988)
1
From the underlying beta distribution, have p=(U-L)L-+L a+B
we
(A.1)
and a/? ~2=(U-L)2~ru+p+l)(a+p)2
(A.21
289 where L = Pt,U= O,,,LL= mean, g2 = variance, and cy and p are parameters of the beta distriand Let bution. r= (P---)l(U-L) Q= ( U-L)'/a2.Then, eqns. (A. 1) and (A.2) can be written as
LY=Q(l-17)I~-(l+(1-)1)/)1)2 C1+(1-V)/?13
(A.3)
P=a(l-rl)lrl
(A.4)