- Email: [email protected]
Analysis of correlated risky cash flow
International
Journal
of Production
Economics,
32 (1993)
269
269-276
Elsevier
Analysis of correlated
risky cash flow
Y.V. Hui”, L.C. Leungb and J.S. Huangc aDepartment
qf Operational Research and Statistics, City Polytechnic qf Hong Kong, Hong Kong b Department qf Operations System Management, Chinese University qf Hong Kong, Hong Kong ‘Industrial and Management Systems Engineering, University of South Florida, Tampa, FL, USA
(Received
12 April 1992; accepted
in revised form 7 May 1993)
Abstract In the analysis of a single cash-flow profile under risk, between-period dependency among cash flows raises considerable difficulty in the evaluation of the profile’s net present value. It is assumed in this paper that the cash-flow profile exhibits a first-order autoregressive time-series structure, with the trend of the AR(l) process being a deterministic AR(i) in itself. The relevant parameters, however, are unknown. It is also assumed that the estimates of the cash-flow in every period are available. Applying an “error-in-variable” analysis on the estimates, the relevant parameters of the time-series model are derived. This then allows the mean and variance of the profile’s NPV to be subsequently evaluated.
1. Introduction In the analysis of a single cash-flow profile under risk, between-period dependency among cash flows represents a major stumbling block in the evaluation of the profile’s net present value (NPV). In the event where perfect correlation between cash flow at periods t and k can be assumed (P~,~ = l), the determination of the variance of the NPV becomes strightforward [l, 21. Where independence can be assumed (P~,~ = 0), the probability density function of the profile’s NPV can be obtained or approximated via a multitude of transform methodologies [3-61. Such transform methodologies can also be used to compute the probability density function of the NPV for cash-flow profiles which exhibit a functional relationship over time, the random parameters being the starting time, the duration, and the initial magnitude of the profile [3]. If the
Correspondence
tional Research Kong,
Hong
to: Y.V. Hui, Department of Operaand Statistics, City Polytechnic of Hong
Kong.
0925-5273/93/$06.00
0
1993 Elsevier
Science
Publishers
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 flow are identically and independently distributed, or they are dependent stationary markovian variates, the NPV distribution would also be approximately normal provided there are “large” number of cash flows [7, pp. 23-291. In general, where between-period dependency exists, other than the aforementioned cases, the NPV distribution cannot be specified. Many works have addressed this dependency problem explicitly. Hiller [2, S] proposed to estimate cash-flow correlations by performing a least-squares linear regression on the cash-flow expectations. Hiller pointed out that the linearity assumption would hold only when the period cash flows formed a multinormal joint distribution. Wagle [9] later both expanded and implemented Hiller’s approach. Assuming a time-series structure, Bussey and Stevens [lo] suggested the use of a maximum likelihood function to estimate the cash-flow correlation; Bonini [ 1 l] identified some dilliculties in the estimation procedure. Bussey and
B.V. All rights
reserved.
Stevens [ 123 later suggested a simulation-regression approach. This approach was conceptually sound. However, Bussey and Stevens had difficulty in generating correlated samples [ 13, 143. In fact, generating dependent variable which are bounded remains a subject of further research [ 15, 161. Other simulation efforts can also be found in [17, 1, 181. Similar to [ 10, 21, Giaccotto [ 193 studied a cash flow profile which was described by a first-order autoregressive time series. Assuming that the oneperiod lag coefficient was a known parameter, the mean and variance of the profile’s NPV was derived; Buck [20] later provided the closed-form expressions for the mean and variance. Modeling cash flow dependency within the time-series framework enables the correlation structure to be explicitly addressed. As such, the problem then lies in determining the appropriate parameter values for the time-series model, and the model’s effectiveness thus depends on the accuracy of the estimation procedure on the available information. Giaccotto [19] suggests that if a new investment has similar characteristics to a project already existing within the firm, previous cash flow data may be used to estimate the unknown parameters; it may also be possible to find a “pureplay” firm which displays cash flow pattern similar to those expected of the new project [21]. For these situations, regression analysis can be performed on the historical data to evaluate the model parameters. However, in the absence of such situations, the analysis will have to rely on expert’s judgement on the period cash flows. But it is inappropriate to perform regression analysis directly on the expert estimates, for they consist of judgement error and are therefore not true realizations of the cash flow profile. In fact, a rigorous approach to overcome these judgement errors appears to be missing in the literature, and it is the purpose of the present work to address this issue. In this paper, we use a statistical methodology to derive information from the expert estimates, which is then utilized to compute the mean and variance of the NPV. Specifi-
tally, we first assume that the dependency structure can be defined by a first-order autoregressive time series. The first-order autoregressive process is Markovian in nature where the response in period t is expressed as the error in period t, u,, plus the product of the lag coefficient 0 and the response in period t - 1. For the process to be stationary, the absolute value of 0 is less than one. When the lag coefficient is close to one, neighboring values in the series have similar values and when the lag coefficient is close to negative one, the series exhibit rapid oscillations. The readers are referred to [22] for the applicability of such a process. Then we propose an “error-in-variable” analysis to relate the timeseries parameters with the expert estimates. This forms the basis for two parameter-estimation procedures, which are both applicable to dependent cash flows. We believe that the present work would fill an important void in the analysis of correlated risky cash flows.
2. Basic model Let Y, be the cash flow at time I, t = 0, 1,. . . , N, and assuming a risk-adjusted discount rate r, the NPV of the cash flow profile is NPV = 2 Y,(l + P))‘. t=o
(1)
Assume that the underlying relationship governing the cash flow profile is a first-order autoregressive time series, i.e., Y,=d,+tIY,_,+u,,
t=l,2
,...,
N,
(2)
where d, is the trend of the profile, t3 is the one-period lag coefficient, 101 < 1 and u, is the random error with E(q) = 0 and Var(a,) = 0’. Here we will further assume that the trend is as follows: d,=rd,_r+/?,
r=l,2
,...,
N,
lxl
Y. V. Hui et al./Ana/ysis
It has been illustrated in [23] that such a/p equation could represent many patterns of cash flow profiles. If a, 3 0, t = 1, 2, . . . , N, Y, will have a deterministic relation and if both 8 and all a,‘~ are zero’s, Y, is simply the trend value. A deterministic trend and homogeneous variance on a, also give the homoskedasticity of Y,. Note that the trend function could in fact consist of certain stochastic parameters; however, this would create substantial difficulty in analyzing the problem and is left for future research. The expected value of the NPV can be expressed as the following closed-form expression (see also the appendix):
E(NPV)
= do
(1 + r)* (1 + r - Q)(l + r - a)
qf‘correlated
271
risk.v cash ,flou
given in [20].
I’(NPV) = Q*
[::=Z)]’
ON (1 + r)N -
- 20(1 + r) [
+
O(1 + r) - 1
6f*yy].
1
1
(5)
Note that eq. (5) does not include parameters of the trend since the trend is deterministic.
N+2
(1 +rL)(B-n)
+(l+r)-N
3. Expert estimates (errors-in-variables)
11 (1 +r)’ +f3 1 8 Nt2
-
(1 + r - O)(Q - a)
r(1 + r -
Q)(l + r - a) CIN+2
(1 +r-C()(B-a)(%-
+“+r)pN
1)
eN+2 -
-
(1 + r - f3)(0 - a)(0 1
1)
’ 11
r(0 - l)(a - 1)
Since the trend value is deterministic, the variance of NPV is identical to that whose cashflow pattern is Y, = 0Y,_, + a,. The methodology to compute the NPV variance, using V( Y,) = 8* V( Y,_ i) + fs* recursively, is shown in [19] and the closed-form value is
From the basic model, if the parameters in (2) and (3) can be properly assessed, the mean and variance of the NPV could be derived. Given that the expert estimates, which consist of judgement errors, are the only available information here, we propose to analyze the expert estimates using an “error-in-variable” approach. Assume that the cash flow sequence but ob(YO,Y1,~. .1 YN} is not available served in terms of (X0,X1, . . . , X,,,) with errors, i.e., Y,=X,+e,,
t=O,l,...,
N,
(6)
where X, is an expert estimate of the true cash flow Y, with an estimate error e,. Here we assume that E(e,) = 0 and I’(e,) = a:. Further we assume that the error in an expert’s judgement in any given period is independent of any other period’s error. We agree that an expert’s judgement error can indeed be correlated over time and it will be explored in future research.
Substituting
4. Weighted least-squares estimation: distribution-free approach
(6) in (2), we have
X, + e, = n, + 0(X,- I + e,- 1) + a,
(7)
or X,-#X,_r-d,=r~,,
t=
1,2 ,...,
N,
(8)
where qt =
(7,-
et +
(9)
8e,-,.
Note that X, is also in the form of a first-order autoregressive time series with trend d, and error qt, where E(qt)=O,
t=
1,2,.
When q,‘s are independent identically distributed with mean 0 and variance g2, we estimate the parameters by minimizing Q’Q which is known as the least-squares estimation. When g,‘s have a covariance matrix V, a weighted least-squares estimation can be used by minimizing q’P’- lg. Here we estimate the unknown parameters by
where
..,N
and Cov(y,,Jj-k)
= Cov(n, -e, 0,-k -cl-k
u2 + a; + 82of_, =
- oTz:_ 1 i
0
+ B&r, + 8e,-k-1)
if k = 0, if k=
1,
if k > 1.
Letting g = (Y(,~_r, . . . , q,)‘, then the covariante matrix of q is
CNXN =
-
(13)
L = V.
(12)
Given the relationships expressed in eqs. (8) and (12), we are now able to design estimation procedures for the model parameters. We suggest two procedures: a distribution-free and a parametric.
x, -
BX(j - dl
f
with v given in eq. (8) and V in eq. (12). Here X,‘s and (T~S are from expert estimates and d, = a + a&-r. Note that distribution assumptions are not required in the weighted least-squares estimation. This estimation procedure applies to a large class of distributions including the gamma distributions, the beta distributions and lognormal distributions.
Y. V. Hui et al./Analysis
5. Maximum likelihood estimation: parametric approach Suppose that a,‘~ and e,‘s are assumed to be normally distributed with distribution N(0, a’) and N(0, cr:), respectively. Then we can employ the maximum likelihood approach in estimating the unknowns. Under normal assumption, q has a joint multinormal distribution with mean 0 and covariance matrix Z:. The likelihood function is L = (2~)-~‘~ IEj1112 exp[( - ~‘Z-‘~)/2)], where Q = X, - 0X,_ 1 - d,. Taking arithm, we have
(14)
the log-
log L= -N log(2n)/2-logl~l/2-~‘~-i~/2. (15) The maximum likelihood estimates of the parameters (0, a, /3, do, 02) are obtained by min
log 1x1 + g’T,-‘q.
(16)
Note that the maximum likelihood method is not restricted to normality assumption. This method applies whenever the joint distribution of q is known. Indeed, the assumption that a,‘s are beta distributed would be a better one; however, it is difficult to obtain the joint distribution of 7 under such an assumption.
6. Implementation One convenient as well as practical way to utilize expert opinions is the PERT approach. Here we request the experts to estimate the pessimistic, modal, and optimistic values of the period cash flows. Then using the underlying beta-variate assumption in PERT, we can approximate the means of the expert estimates for each period. These means, in turn, are assumed to be {X,, X1,. . . , X,}, estimates of the true cash flows with an error component. Moreover, the variance of expert estimates are assumed to be the variance of the error term e,,
qf’correlated
273
risky cash,fiow
i.e., 0:. It should be pointed out that since the variance a:‘s are estimated from expert estimates, it implies that 0;‘s are stochastic variables. As an example, we consider the expert estimates for a ten-period profile given in Bussey [12], where the corresponding means (X,‘s) and variances (cr:‘s) are approximated using the beta-variate assumption. Assuming that the true cash flow profile is characterized by eqs. (2) and (3), and that the risk-adjusted discount rate is 10%. we can proceed with estimations of the time-series model parameters 0, do, r, fl, and a2. Since the normality assumption is inappropriate, we use the weighted least-squares method: minimize q’ V- ‘7 as given by eq. (13). Also as it is very cumbersome to obtain the closed-form for q’V_ ‘q, the cyclic coordinate method [23], a multidimensional search algorithm without using derivatives, is used to search for the optimal solution. It is noted that as the number of periods in eq. (13) increases, the complexity of the objective function increases. In the optimization algorithm, instead of expanding eq. (13), we calculate each element in the q vector and I/ matrix, followed by matrix manipulation to obtain the value of the objective function. Because of the complexity of the objective function, there may exit multiple local minimums; the convergence to a local solution may be very sensitive to the chosen starting point and that the function may explode at some points. Several starting points were used in the minimum searching process. The corresponding solution was found as e=
0.3955,
d^= - 33,475.50,
&=0.5451,
/?= 8014.88,
22 = 1.0937( 107)
E^(NPV)=32251.10,
p(NPV)=
and 1.0389(10*).
It should be pointed out that since the objective function does not well behave, the above solution cannot be guaranteed as the global minimum.
If the maximum likelihood estimation approach is to be used, the joint distribution of q would have to be determined. As indicated earlier, when a,‘~ are beta distributed, this joint distribution is difficult to obtain and hence creates problem in assessing the likelihood function. Should normal assumption be applicable to the cash flow profile, we could then utilize eq. (16) to estimate 19; the mean and variance of the NPV could subsequently be computed.
7. Bounded dependent cash flows Within the time-series framework, the property of the error term a, has a major impact on the choice of an appropriate estimation procedure. While efficient estimation procedure can be derived if a, is assumed to be N(0, a’), however, the normality assumption could be unrealistic for cash flow problems. For, if the a,‘~ are normal variates, the period cash flow Y,‘s are implicitly normal variates as well. This can be seen in eq. (2), i.e., Y, = d, + 0Y,_ 1 + a,, where u, is a linear combination of Y,_ 1 and Y,. Since a normal variate is unbounded, and since cash flow are bounded in most realistic situations, the normality assumption is at best a poor one. In the more realistic situation where the cash flows are both dependent and bounded, it is then necessary that the estimation procedure be applicable for bounded a,; this is due to the similar argument that if the Y,‘s are bounded, the a,‘s would have to be bounded as well. This particular aspect in the treatment of risky cash flows appears to have received little attention, and we believe that the proposed methodology does incorporate this situation.
lysis, we are able to propose two estimation procedures which can be applicable to bounded or unbounded a,‘~. That is, our two procedures are applicable to bounded or unbounded cash flow whose correlation structure is defined by the time-series model. To illustrate how our approach can be applicable to bounded dependent cash flows, an example profile of beta variates is used. It should be pointed out that while the weighted leastsquares approach is very general since distribution assumptions are not required, it is not as precise as the maximum likelihood approach. Unfortunately, the maximum likelihood approach can be very difficult to implement since the joint distribution of the cash flows is required. This will be left for future research. Another extension to the present work would be the analysis of correlation effects across different cash-flow profiles.
9. Appendix 9.1. Expected
value
qf NPV
Let ft be the future worth of the cash flow profile at time t, we write the following system of difference equation:
EU)=(l +r)E(.L1)+~E(yr- 1)+&l
B=B or
8. Summary In this paper, after expressing 8 and a, of the time-series model in terms of the known information using an “error-in-variable” ana-
where
ur = CE(ft),WY,), 4, PI’
+/A
Y. V. Hui et al./Anuiysis
of correlated risky cash ,flm
and
-
l+r A=
f3 a
1
0
6a
1
0
0
X 1
0
0
0
275
Of+2[(1+
+ E”2[(t
r -
O)(O -
a)(0
-
l)]-’
+ r - ct)(8 - !$(a - l)]_’
- [r(0 - I)@ - 1)1-l} and
1
Here we factorize A into PAP- ‘, where P is the matrix whose column vectors are eigenvectors of A and A is the matrix with eigenvalues of A across the main diagonal. We have
E(y,)=~o(8’f1-x’+‘)(8-cr)+~B’+~(GI-l) - cl’+‘(tj - 1) + (0 - Gl)f/ C(@- @I(@- l)(a - l)], with
P=
rl0 -
o(l+r-or
srqi+r-r)-‘(n-cY-’
L
A=
p-1 =
0
Since E(J) represents the expected future worth of the cash flow profile at time t, the corresponding expected net present value, as it is given in eq. (4), can be obtained by multiplying E(f,) with a discount factor (1 f Y)-‘.
,1
0 0
.
d, = a’& + /3(c? - l)/(a - 1).
-r-yn-r)-‘(r-l)-‘l
0
111 + r)
0
I
0
0
References Cl1 Fuller,
(1 4 r)Z
(cl -
I)-’ 1
0
with u. = [do, do, do, PI’. It follows that E(.f,)=d,((l
+r)‘+2(1 +r-@)-l(l
+r-cr)-l
- 13’+2(1+ r - 0)_‘(8 - a)-’ + ~2’+~(1+ +P((t
+r)‘y(l
r -
a)-‘(8
-
+r-@)(l
cc-‘}
+r-a)-’
1
R.J. and Kim, S.H., 1980, Inter-temporal correlation of cash flow and the risk of multi-period investment projects. J. Financial and Quant. Anal., XV(5): f 149-- 1162. c21Hiller, F.S., 1963, The derivation of probabilistic information for the evaluation of risky investments. Mgmt. Sci., 7(3): 443-457. c31 Barnes, J.W., Zinn, CD. and Eldred, B.S., 1981, A methodology for obtaining the probability density function of the present worth of probabilistic cash flow profiles. AIIE Trans.. lO(3): 226 ~236. c41Tanchoco, J.M.A. and Buck, J.R., 1971. A closedform methodology for computing present worth statistics of risky discrete cash-flows. ATIE Trans., 9(3): 278-287. L.E., 1975, Expected c51Young, D. and Contreras, present worth of cash flows under uncertain timing. The Eng. Economist, 20(4): 257 -268. El Zinn, C.D. and Lesso, W.G., 1977, A probabilistic approach to risk analysis in capital investment projects. The Eng. Economist, 22(4): 239-260. PI Hiller, F.S., 1969, The Evaluation of Risky Interrelated Investments. North-Holland, Amsterdam. PI Hiller, F.S., 1965, Supplement to the derivation of probabilistic information for the evaluation of risky investments. Mgmt. Sci., 1l(3): 485 -487.
1~
276
Y. V. Hui et ul./Analysis
[9] Wagle, B., 1967, A statistical analysis of risk in capital investment projects. Oper. Res. Quart., 18(l): 13-33. [lo] Bussey, L.E. and Stevens, G.T. Jr., 1972, Formulating correlated cash flow streams. The Eng. Economist, 18(l): l-31. [1 l] Bonini, C.P., 1975. Comment on ‘formulating correlated cash flow stream’. The Eng. Economist, 20(3): 209 -214. [ 121 Bussey, L.E., 1978, The Economic Analysis of Industrial Projects. Prentice-Hall, Englewood Cliffs, NJ. 1131 Bussey, L.E., 1981, In response to a book review. The Eng. Economist, 26(2): 76678. [14] Hodder, J.E. and Jucker, J.V., 1980, Book Review: The Economic Analysis of Industrial Projects. Prentice-Hall, Englewood Cliffs, NJ; L.E. Bussey, 1978, The Eng. Economist, 25(2): 146-147. [15] McKenzie, E., 1985, An autoregressive process for beta random variables. Mgmt. Sci., 31(S): 988-997. 1161 Schmeiser, B. and Lal, R., 1980, Multivariate modeling in simulation: A survey. In Proc. 33rd Annual Technical Conference Transactions. American Society for Quality Control Atlanta, p. 2522261. [17] Bussey, L.E. and Stevens, G.T. Jr., 1971, Net present value from complex cash flow streams by simulation. AIIE Trans., III( 1): 8 l-89.
of‘ correlated
risk.’ cash ,jlm~
[18] Kryzanowski, L., Lusztig. P. and Schwab, B., 1972, Monte Carlo simulation and capital expenditure decisions -- a case study. The Eng. Economist, 18( 1): 31-57. [19] Giaccotto, C., 1984, A simplified approach to risk analysis in capital budgeting with serially correlated cash flows. The Eng. Economist, 29(4): 273-286. [20] Buck, J.R., 1984, A technical note and comments on Carmelo Giaccotto’s a simplified approach to risk analysis in capital budgeting with serially correlated cash flows’. The Eng. Economist, 29(4): 2733286; Working Paper, Department of Industrial Engineering and Management Engineering, University of Iowa. 1211 Fuller, R.J. and Kerr, H.S., 1981. Estimating the divisional cost of capital: an analysis of the pure-play technique. The J. Finance, XXXVI: 997- 1009. 1221 Box, G.E.P. and Jenkins, G.M., 1976, Time Series Analysis: Forecasting and Control. Holden-Day, Oakland, CA. [23] Buck, J.R. and Hill, T.W. Jr., 1975. Alpha/beta difference equations and their zeta transforms in economic analysis. AIIE Trans., 7(3): 330-358. [24] Bazaraa, MS. and Shetty, C.M., 1979, Nonlinear Programming: Theory and Algorithms. Wiley, New York.
Journal
of Production
Economics,
32 (1993)
269
269-276
Elsevier
Analysis of correlated
risky cash flow
Y.V. Hui”, L.C. Leungb and J.S. Huangc aDepartment
qf Operational Research and Statistics, City Polytechnic qf Hong Kong, Hong Kong b Department qf Operations System Management, Chinese University qf Hong Kong, Hong Kong ‘Industrial and Management Systems Engineering, University of South Florida, Tampa, FL, USA
(Received
12 April 1992; accepted
in revised form 7 May 1993)
Abstract In the analysis of a single cash-flow profile under risk, between-period dependency among cash flows raises considerable difficulty in the evaluation of the profile’s net present value. It is assumed in this paper that the cash-flow profile exhibits a first-order autoregressive time-series structure, with the trend of the AR(l) process being a deterministic AR(i) in itself. The relevant parameters, however, are unknown. It is also assumed that the estimates of the cash-flow in every period are available. Applying an “error-in-variable” analysis on the estimates, the relevant parameters of the time-series model are derived. This then allows the mean and variance of the profile’s NPV to be subsequently evaluated.
1. Introduction In the analysis of a single cash-flow profile under risk, between-period dependency among cash flows represents a major stumbling block in the evaluation of the profile’s net present value (NPV). In the event where perfect correlation between cash flow at periods t and k can be assumed (P~,~ = l), the determination of the variance of the NPV becomes strightforward [l, 21. Where independence can be assumed (P~,~ = 0), the probability density function of the profile’s NPV can be obtained or approximated via a multitude of transform methodologies [3-61. Such transform methodologies can also be used to compute the probability density function of the NPV for cash-flow profiles which exhibit a functional relationship over time, the random parameters being the starting time, the duration, and the initial magnitude of the profile [3]. If the
Correspondence
tional Research Kong,
Hong
to: Y.V. Hui, Department of Operaand Statistics, City Polytechnic of Hong
Kong.
0925-5273/93/$06.00
0
1993 Elsevier
Science
Publishers
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 flow are identically and independently distributed, or they are dependent stationary markovian variates, the NPV distribution would also be approximately normal provided there are “large” number of cash flows [7, pp. 23-291. In general, where between-period dependency exists, other than the aforementioned cases, the NPV distribution cannot be specified. Many works have addressed this dependency problem explicitly. Hiller [2, S] proposed to estimate cash-flow correlations by performing a least-squares linear regression on the cash-flow expectations. Hiller pointed out that the linearity assumption would hold only when the period cash flows formed a multinormal joint distribution. Wagle [9] later both expanded and implemented Hiller’s approach. Assuming a time-series structure, Bussey and Stevens [lo] suggested the use of a maximum likelihood function to estimate the cash-flow correlation; Bonini [ 1 l] identified some dilliculties in the estimation procedure. Bussey and
B.V. All rights
reserved.
Stevens [ 123 later suggested a simulation-regression approach. This approach was conceptually sound. However, Bussey and Stevens had difficulty in generating correlated samples [ 13, 143. In fact, generating dependent variable which are bounded remains a subject of further research [ 15, 161. Other simulation efforts can also be found in [17, 1, 181. Similar to [ 10, 21, Giaccotto [ 193 studied a cash flow profile which was described by a first-order autoregressive time series. Assuming that the oneperiod lag coefficient was a known parameter, the mean and variance of the profile’s NPV was derived; Buck [20] later provided the closed-form expressions for the mean and variance. Modeling cash flow dependency within the time-series framework enables the correlation structure to be explicitly addressed. As such, the problem then lies in determining the appropriate parameter values for the time-series model, and the model’s effectiveness thus depends on the accuracy of the estimation procedure on the available information. Giaccotto [19] suggests that if a new investment has similar characteristics to a project already existing within the firm, previous cash flow data may be used to estimate the unknown parameters; it may also be possible to find a “pureplay” firm which displays cash flow pattern similar to those expected of the new project [21]. For these situations, regression analysis can be performed on the historical data to evaluate the model parameters. However, in the absence of such situations, the analysis will have to rely on expert’s judgement on the period cash flows. But it is inappropriate to perform regression analysis directly on the expert estimates, for they consist of judgement error and are therefore not true realizations of the cash flow profile. In fact, a rigorous approach to overcome these judgement errors appears to be missing in the literature, and it is the purpose of the present work to address this issue. In this paper, we use a statistical methodology to derive information from the expert estimates, which is then utilized to compute the mean and variance of the NPV. Specifi-
tally, we first assume that the dependency structure can be defined by a first-order autoregressive time series. The first-order autoregressive process is Markovian in nature where the response in period t is expressed as the error in period t, u,, plus the product of the lag coefficient 0 and the response in period t - 1. For the process to be stationary, the absolute value of 0 is less than one. When the lag coefficient is close to one, neighboring values in the series have similar values and when the lag coefficient is close to negative one, the series exhibit rapid oscillations. The readers are referred to [22] for the applicability of such a process. Then we propose an “error-in-variable” analysis to relate the timeseries parameters with the expert estimates. This forms the basis for two parameter-estimation procedures, which are both applicable to dependent cash flows. We believe that the present work would fill an important void in the analysis of correlated risky cash flows.
2. Basic model Let Y, be the cash flow at time I, t = 0, 1,. . . , N, and assuming a risk-adjusted discount rate r, the NPV of the cash flow profile is NPV = 2 Y,(l + P))‘. t=o
(1)
Assume that the underlying relationship governing the cash flow profile is a first-order autoregressive time series, i.e., Y,=d,+tIY,_,+u,,
t=l,2
,...,
N,
(2)
where d, is the trend of the profile, t3 is the one-period lag coefficient, 101 < 1 and u, is the random error with E(q) = 0 and Var(a,) = 0’. Here we will further assume that the trend is as follows: d,=rd,_r+/?,
r=l,2
,...,
N,
lxl
Y. V. Hui et al./Ana/ysis
It has been illustrated in [23] that such a/p equation could represent many patterns of cash flow profiles. If a, 3 0, t = 1, 2, . . . , N, Y, will have a deterministic relation and if both 8 and all a,‘~ are zero’s, Y, is simply the trend value. A deterministic trend and homogeneous variance on a, also give the homoskedasticity of Y,. Note that the trend function could in fact consist of certain stochastic parameters; however, this would create substantial difficulty in analyzing the problem and is left for future research. The expected value of the NPV can be expressed as the following closed-form expression (see also the appendix):
E(NPV)
= do
(1 + r)* (1 + r - Q)(l + r - a)
qf‘correlated
271
risk.v cash ,flou
given in [20].
I’(NPV) = Q*
[::=Z)]’
ON (1 + r)N -
- 20(1 + r) [
+
O(1 + r) - 1
6f*yy].
1
1
(5)
Note that eq. (5) does not include parameters of the trend since the trend is deterministic.
N+2
(1 +rL)(B-n)
+(l+r)-N
3. Expert estimates (errors-in-variables)
11 (1 +r)’ +f3 1 8 Nt2
-
(1 + r - O)(Q - a)
r(1 + r -
Q)(l + r - a) CIN+2
(1 +r-C()(B-a)(%-
+“+r)pN
1)
eN+2 -
-
(1 + r - f3)(0 - a)(0 1
1)
’ 11
r(0 - l)(a - 1)
Since the trend value is deterministic, the variance of NPV is identical to that whose cashflow pattern is Y, = 0Y,_, + a,. The methodology to compute the NPV variance, using V( Y,) = 8* V( Y,_ i) + fs* recursively, is shown in [19] and the closed-form value is
From the basic model, if the parameters in (2) and (3) can be properly assessed, the mean and variance of the NPV could be derived. Given that the expert estimates, which consist of judgement errors, are the only available information here, we propose to analyze the expert estimates using an “error-in-variable” approach. Assume that the cash flow sequence but ob(YO,Y1,~. .1 YN} is not available served in terms of (X0,X1, . . . , X,,,) with errors, i.e., Y,=X,+e,,
t=O,l,...,
N,
(6)
where X, is an expert estimate of the true cash flow Y, with an estimate error e,. Here we assume that E(e,) = 0 and I’(e,) = a:. Further we assume that the error in an expert’s judgement in any given period is independent of any other period’s error. We agree that an expert’s judgement error can indeed be correlated over time and it will be explored in future research.
Substituting
4. Weighted least-squares estimation: distribution-free approach
(6) in (2), we have
X, + e, = n, + 0(X,- I + e,- 1) + a,
(7)
or X,-#X,_r-d,=r~,,
t=
1,2 ,...,
N,
(8)
where qt =
(7,-
et +
(9)
8e,-,.
Note that X, is also in the form of a first-order autoregressive time series with trend d, and error qt, where E(qt)=O,
t=
1,2,.
When q,‘s are independent identically distributed with mean 0 and variance g2, we estimate the parameters by minimizing Q’Q which is known as the least-squares estimation. When g,‘s have a covariance matrix V, a weighted least-squares estimation can be used by minimizing q’P’- lg. Here we estimate the unknown parameters by
where
..,N
and Cov(y,,Jj-k)
= Cov(n, -e, 0,-k -cl-k
u2 + a; + 82of_, =
- oTz:_ 1 i
0
+ B&r, + 8e,-k-1)
if k = 0, if k=
1,
if k > 1.
Letting g = (Y(,~_r, . . . , q,)‘, then the covariante matrix of q is
CNXN =
-
(13)
L = V.
(12)
Given the relationships expressed in eqs. (8) and (12), we are now able to design estimation procedures for the model parameters. We suggest two procedures: a distribution-free and a parametric.
x, -
BX(j - dl
f
with v given in eq. (8) and V in eq. (12). Here X,‘s and (T~S are from expert estimates and d, = a + a&-r. Note that distribution assumptions are not required in the weighted least-squares estimation. This estimation procedure applies to a large class of distributions including the gamma distributions, the beta distributions and lognormal distributions.
Y. V. Hui et al./Analysis
5. Maximum likelihood estimation: parametric approach Suppose that a,‘~ and e,‘s are assumed to be normally distributed with distribution N(0, a’) and N(0, cr:), respectively. Then we can employ the maximum likelihood approach in estimating the unknowns. Under normal assumption, q has a joint multinormal distribution with mean 0 and covariance matrix Z:. The likelihood function is L = (2~)-~‘~ IEj1112 exp[( - ~‘Z-‘~)/2)], where Q = X, - 0X,_ 1 - d,. Taking arithm, we have
(14)
the log-
log L= -N log(2n)/2-logl~l/2-~‘~-i~/2. (15) The maximum likelihood estimates of the parameters (0, a, /3, do, 02) are obtained by min
log 1x1 + g’T,-‘q.
(16)
Note that the maximum likelihood method is not restricted to normality assumption. This method applies whenever the joint distribution of q is known. Indeed, the assumption that a,‘s are beta distributed would be a better one; however, it is difficult to obtain the joint distribution of 7 under such an assumption.
6. Implementation One convenient as well as practical way to utilize expert opinions is the PERT approach. Here we request the experts to estimate the pessimistic, modal, and optimistic values of the period cash flows. Then using the underlying beta-variate assumption in PERT, we can approximate the means of the expert estimates for each period. These means, in turn, are assumed to be {X,, X1,. . . , X,}, estimates of the true cash flows with an error component. Moreover, the variance of expert estimates are assumed to be the variance of the error term e,,
qf’correlated
273
risky cash,fiow
i.e., 0:. It should be pointed out that since the variance a:‘s are estimated from expert estimates, it implies that 0;‘s are stochastic variables. As an example, we consider the expert estimates for a ten-period profile given in Bussey [12], where the corresponding means (X,‘s) and variances (cr:‘s) are approximated using the beta-variate assumption. Assuming that the true cash flow profile is characterized by eqs. (2) and (3), and that the risk-adjusted discount rate is 10%. we can proceed with estimations of the time-series model parameters 0, do, r, fl, and a2. Since the normality assumption is inappropriate, we use the weighted least-squares method: minimize q’ V- ‘7 as given by eq. (13). Also as it is very cumbersome to obtain the closed-form for q’V_ ‘q, the cyclic coordinate method [23], a multidimensional search algorithm without using derivatives, is used to search for the optimal solution. It is noted that as the number of periods in eq. (13) increases, the complexity of the objective function increases. In the optimization algorithm, instead of expanding eq. (13), we calculate each element in the q vector and I/ matrix, followed by matrix manipulation to obtain the value of the objective function. Because of the complexity of the objective function, there may exit multiple local minimums; the convergence to a local solution may be very sensitive to the chosen starting point and that the function may explode at some points. Several starting points were used in the minimum searching process. The corresponding solution was found as e=
0.3955,
d^= - 33,475.50,
&=0.5451,
/?= 8014.88,
22 = 1.0937( 107)
E^(NPV)=32251.10,
p(NPV)=
and 1.0389(10*).
It should be pointed out that since the objective function does not well behave, the above solution cannot be guaranteed as the global minimum.
If the maximum likelihood estimation approach is to be used, the joint distribution of q would have to be determined. As indicated earlier, when a,‘~ are beta distributed, this joint distribution is difficult to obtain and hence creates problem in assessing the likelihood function. Should normal assumption be applicable to the cash flow profile, we could then utilize eq. (16) to estimate 19; the mean and variance of the NPV could subsequently be computed.
7. Bounded dependent cash flows Within the time-series framework, the property of the error term a, has a major impact on the choice of an appropriate estimation procedure. While efficient estimation procedure can be derived if a, is assumed to be N(0, a’), however, the normality assumption could be unrealistic for cash flow problems. For, if the a,‘~ are normal variates, the period cash flow Y,‘s are implicitly normal variates as well. This can be seen in eq. (2), i.e., Y, = d, + 0Y,_ 1 + a,, where u, is a linear combination of Y,_ 1 and Y,. Since a normal variate is unbounded, and since cash flow are bounded in most realistic situations, the normality assumption is at best a poor one. In the more realistic situation where the cash flows are both dependent and bounded, it is then necessary that the estimation procedure be applicable for bounded a,; this is due to the similar argument that if the Y,‘s are bounded, the a,‘s would have to be bounded as well. This particular aspect in the treatment of risky cash flows appears to have received little attention, and we believe that the proposed methodology does incorporate this situation.
lysis, we are able to propose two estimation procedures which can be applicable to bounded or unbounded a,‘~. That is, our two procedures are applicable to bounded or unbounded cash flow whose correlation structure is defined by the time-series model. To illustrate how our approach can be applicable to bounded dependent cash flows, an example profile of beta variates is used. It should be pointed out that while the weighted leastsquares approach is very general since distribution assumptions are not required, it is not as precise as the maximum likelihood approach. Unfortunately, the maximum likelihood approach can be very difficult to implement since the joint distribution of the cash flows is required. This will be left for future research. Another extension to the present work would be the analysis of correlation effects across different cash-flow profiles.
9. Appendix 9.1. Expected
value
qf NPV
Let ft be the future worth of the cash flow profile at time t, we write the following system of difference equation:
EU)=(l +r)E(.L1)+~E(yr- 1)+&l
B=B or
8. Summary In this paper, after expressing 8 and a, of the time-series model in terms of the known information using an “error-in-variable” ana-
where
ur = CE(ft),WY,), 4, PI’
+/A
Y. V. Hui et al./Anuiysis
of correlated risky cash ,flm
and
-
l+r A=
f3 a
1
0
6a
1
0
0
X 1
0
0
0
275
Of+2[(1+
+ E”2[(t
r -
O)(O -
a)(0
-
l)]-’
+ r - ct)(8 - !$(a - l)]_’
- [r(0 - I)@ - 1)1-l} and
1
Here we factorize A into PAP- ‘, where P is the matrix whose column vectors are eigenvectors of A and A is the matrix with eigenvalues of A across the main diagonal. We have
E(y,)=~o(8’f1-x’+‘)(8-cr)+~B’+~(GI-l) - cl’+‘(tj - 1) + (0 - Gl)f/ C(@- @I(@- l)(a - l)], with
P=
rl0 -
o(l+r-or
srqi+r-r)-‘(n-cY-’
L
A=
p-1 =
0
Since E(J) represents the expected future worth of the cash flow profile at time t, the corresponding expected net present value, as it is given in eq. (4), can be obtained by multiplying E(f,) with a discount factor (1 f Y)-‘.
,1
0 0
.
d, = a’& + /3(c? - l)/(a - 1).
-r-yn-r)-‘(r-l)-‘l
0
111 + r)
0
I
0
0
References Cl1 Fuller,
(1 4 r)Z
(cl -
I)-’ 1
0
with u. = [do, do, do, PI’. It follows that E(.f,)=d,((l
+r)‘+2(1 +r-@)-l(l
+r-cr)-l
- 13’+2(1+ r - 0)_‘(8 - a)-’ + ~2’+~(1+ +P((t
+r)‘y(l
r -
a)-‘(8
-
+r-@)(l
cc-‘}
+r-a)-’
1
R.J. and Kim, S.H., 1980, Inter-temporal correlation of cash flow and the risk of multi-period investment projects. J. Financial and Quant. Anal., XV(5): f 149-- 1162. c21Hiller, F.S., 1963, The derivation of probabilistic information for the evaluation of risky investments. Mgmt. Sci., 7(3): 443-457. c31 Barnes, J.W., Zinn, CD. and Eldred, B.S., 1981, A methodology for obtaining the probability density function of the present worth of probabilistic cash flow profiles. AIIE Trans.. lO(3): 226 ~236. c41Tanchoco, J.M.A. and Buck, J.R., 1971. A closedform methodology for computing present worth statistics of risky discrete cash-flows. ATIE Trans., 9(3): 278-287. L.E., 1975, Expected c51Young, D. and Contreras, present worth of cash flows under uncertain timing. The Eng. Economist, 20(4): 257 -268. El Zinn, C.D. and Lesso, W.G., 1977, A probabilistic approach to risk analysis in capital investment projects. The Eng. Economist, 22(4): 239-260. PI Hiller, F.S., 1969, The Evaluation of Risky Interrelated Investments. North-Holland, Amsterdam. PI Hiller, F.S., 1965, Supplement to the derivation of probabilistic information for the evaluation of risky investments. Mgmt. Sci., 1l(3): 485 -487.
1~
276
Y. V. Hui et ul./Analysis
[9] Wagle, B., 1967, A statistical analysis of risk in capital investment projects. Oper. Res. Quart., 18(l): 13-33. [lo] Bussey, L.E. and Stevens, G.T. Jr., 1972, Formulating correlated cash flow streams. The Eng. Economist, 18(l): l-31. [1 l] Bonini, C.P., 1975. Comment on ‘formulating correlated cash flow stream’. The Eng. Economist, 20(3): 209 -214. [ 121 Bussey, L.E., 1978, The Economic Analysis of Industrial Projects. Prentice-Hall, Englewood Cliffs, NJ. 1131 Bussey, L.E., 1981, In response to a book review. The Eng. Economist, 26(2): 76678. [14] Hodder, J.E. and Jucker, J.V., 1980, Book Review: The Economic Analysis of Industrial Projects. Prentice-Hall, Englewood Cliffs, NJ; L.E. Bussey, 1978, The Eng. Economist, 25(2): 146-147. [15] McKenzie, E., 1985, An autoregressive process for beta random variables. Mgmt. Sci., 31(S): 988-997. 1161 Schmeiser, B. and Lal, R., 1980, Multivariate modeling in simulation: A survey. In Proc. 33rd Annual Technical Conference Transactions. American Society for Quality Control Atlanta, p. 2522261. [17] Bussey, L.E. and Stevens, G.T. Jr., 1971, Net present value from complex cash flow streams by simulation. AIIE Trans., III( 1): 8 l-89.
of‘ correlated
risk.’ cash ,jlm~
[18] Kryzanowski, L., Lusztig. P. and Schwab, B., 1972, Monte Carlo simulation and capital expenditure decisions -- a case study. The Eng. Economist, 18( 1): 31-57. [19] Giaccotto, C., 1984, A simplified approach to risk analysis in capital budgeting with serially correlated cash flows. The Eng. Economist, 29(4): 273-286. [20] Buck, J.R., 1984, A technical note and comments on Carmelo Giaccotto’s a simplified approach to risk analysis in capital budgeting with serially correlated cash flows’. The Eng. Economist, 29(4): 2733286; Working Paper, Department of Industrial Engineering and Management Engineering, University of Iowa. 1211 Fuller, R.J. and Kerr, H.S., 1981. Estimating the divisional cost of capital: an analysis of the pure-play technique. The J. Finance, XXXVI: 997- 1009. 1221 Box, G.E.P. and Jenkins, G.M., 1976, Time Series Analysis: Forecasting and Control. Holden-Day, Oakland, CA. [23] Buck, J.R. and Hill, T.W. Jr., 1975. Alpha/beta difference equations and their zeta transforms in economic analysis. AIIE Trans., 7(3): 330-358. [24] Bazaraa, MS. and Shetty, C.M., 1979, Nonlinear Programming: Theory and Algorithms. Wiley, New York.