Evaluation of risk in investment alternatives

Comput. & Indus. Engng, Vo]. I. pp. 185-197. Pergamon Press. 1977. Printed in Great Britain

E V A L U A T I O N O F R I S K IN I N V E S T M E N T A L T E R N A T I V E S M. SADEK EID Universit6 de Moncton, Moncton, N.B., Canada

and H. K. ELDIN Oklahoma State University, Stillwater, Oklahoma, U.S.A.

PAPER PRESENTED TO THE FIRST NATIONAL CONFERENCE ON COMPUTERS AND INDUSTRIAL ENGINEERING, TULSA, OKLAHOMA, JUNE 2-4, 1976 (Received 1 November 1976) AMtraet--Economic decisions regarding capital investments involve choosing different proposed alternatives. Each of these alternatives includes events and outcomes that would occur in the future. In making such decisions, there is always an amount of risk involved regarding the future outcomes of the different alternatives. The amount of risk involved is an important consideration in the evaluation of the proposed investments. Neglecting the risk may result in faulty decisions. Furthermore, risk information permits management to weigh more precisely the possible consequences of the proposed investments. An integrated system has been developed and can be used for evaluating risk in investments. The paper will give the mathematical background and a description of the simulation system. Utility P rogram For the Analysis of Risk (UPFAR) is an integrated system composed of four modules: the Probability D/stributinn Fitting Program (PRODIP) module which fits probability distributions to empirical data; the General Utility Simulation System (GUSS) module which is a general purpose simulator; the Investment Problem Model Routine (INMO) module which is a general investment analysis model; and the Routine for Utility Measurement (RUM) module which evaluates the utility of an investment. UPFAR provides a tool for the analysis of risk in investments. UPFAR is easy to use, can be applied to a wide range of risk analysis problems, and can be installed in the majority of computer installations. 1. I N T R O D U C T I O N

Risk is an inherent factor in most management decisions. As the decision gets more complex with interacting decision parameters, uncertainties are compounded increasing the difficulty of decision making. Risk in projects was recognized by conventional management in their decision making process. Lacking models to account for risk, investment problems were dealt with by initially assuming deterministic data, based on the expected value or the most likely outcome for example, and then augmenting the analysis by: • Using a shorter payout period to recover the initial investment. • Using a higher rate of return than is normally accepted. • Using conservative estimates of the parameters including uncertainties. Such heuristic approaches may however be misleading and can lead to faulty decisions. Furthermore, scientific management normally relies on quantitative models that can effectively utilize available data to provide decision making norms. Mathematical models and simulation techniques can be used to evaluate a measure for risk in investment. Mathematical models have been developed which utilize higher mathematics and theory of probability. However, such models are normally either highly complex for implementation or include assumptions and approximations to simplify the model, but that would also result in final solutions that are not repreSentative of the original problem. Simulation models can be effective when analytical solutions are not possible. This paper offers an integrated simulation system for the evaluation of an investment opportunity that is subject to risk. A risk analysis study includes the following phases: • Evaluating the probability distributions of the investment components. 185

186

M. SADEK EID and H. K. Elolr~

• A simulation model to study the interaction of the stochastic outcomes. • Measures of utility for evaluating the risks that can be used for the investment selection.

2. RISK A N A L Y S E S C O N C E P T S

2.1. Risk measurement When the parameters of an investment are all deterministic, one can use any of different formulae, developed in engineering economy references, to evaluate its worthiness. Similar formulae were developed by Reisman and Rao [25] for some special stochastic variations of the problem. To evaluate the risk in an investment, the following general model can be used: NPV

Y,(1 + r)-'

(1)

where: NPV = Net Present Worth of investment, n = life of project, r = company's minimum acceptable rate of return, I:, = vector of cash flows at period t, = [YE],, k = 1, 2 . . . . . m, and m = number of different types of cash flows. As an example of cash flow types for model (1), y, can be investment in land, y2 working capital, y5 investment in a certain type of equipment, y,o revenue from certain sales, etc. A cash flow, YE, does not necessarily have to be a simple random variable, but can be the resultant of the interaction of different components, compounded together with simple or complex relationships. These components can assume deterministic or stochastic variables, dependent or independent. The life of the project and the rate of return of the company may also be assumed as random variables. As a result of these possible stochastic parameters included in the NPV calculation, model 1, the resulting NPV is also a random variable. The distribution of the NPV is dependent on the distribution of its component elements. Assumptions have been made by different researchers that the NPV distribution is normal. Such assumptions can sometimes be correct. However, the normality assumption is not generally true as in Horowitz's model[18], and hence the actual distribution of NPV need to be determined in a risk analysis study which is made possible by the model presented here. 2.2. Probability distributions A major task in risk analysis is the assignment of probabilities and distributions to the different parameters of the investment problem. Appropriate considerations should be given to the physical processes that govern the properties of a given distribution function before assigning it to a stochastic variable, e.g. continuous, discrete, etc. Three approaches could be used for assigning probabilities, depending on the problem: (a) The shape of the probability distribution of the stochastic variable is assumed based on previous experiences or speculations about the variable. The statistical parameters of the assumed distribution are also assumed in the same manner. (b) Subjective probabilities may be estimated or assumed based on previous experiences or speculations about the variable. These subjective probabilities can be used discretely, interpolated for simulating the continuous case, or can be used as the basis for a maximum entropy distribution assumption[28]. (c) Where empirical or historical data are available, these data can be used to determine the frequency or probability distribution of the stochastic variable under study. Three methods are available to generate a distribution for a stochastic variable based on the available information.

Evaluation of risk in investmentalternatives

187

(1) Maximum entropy distributions [28]. Based on Jaynes' principle of maximum entropy, the minimally prejudiced probability distribution is that which maximizes the function: ~p = - ~

p, In p,.

(2)

i

where q~ is the entropy and pl is the probability of outcome i. Now let us denot: pk = probability of xk and E(x)= expected value of x. The maximum entropy formalism gives the different distributions that fit the available information. (a) Using only the information that

~

Pk = 1

(3)

k~O

the uni/orm distribution would be the minimally prejudiced distribution for this information. (b) Given the information that

~

pk = 1

(4)

p~x~ = E ( x )

(5)

k=O

and k=O

the Exponential distribution is then the minimally prejudiced distribution for this information. (c) Given the information that

~ Pk = 1,

(6)

k=O

and

pkxk = E(x),

(7)

p~x~~ = E(x~).

(8)

k=O

k=O

The Truncated Gaussian distribution is the minimally prejudiced for this information. In the case when the range of x is - ~ - < x-< ~, the distribution in this case is the Normal Gaussian. (d) Given the information that k=O

Pk = 1

f ~ p~x~ = E ( x ) k=O

' ~ P k lnxk = E(ln x).

k=O

(9)

(10) (I1)

(e) Given the information that O--
(12)

~pk=l,

(13)

k=O

~ P k In xk = E(ln x), k=o

~ p ~ In (1 - xk) = E(ln (1 - x)).

k=O

The Beta distribution is the minimally prejudiced distribution for this information.

(14)

05)

188

M. SADEK EID and H. K. ELDIN

(f) Given the information that

~ Pk

= 1

(16)

i=l

' • Pin k= E(ln xk) XE

p~x/' = E(xk~).

(17) (18)

i=l

This leads to a family of distributions of which the Weibull distribution is a special case. (2) Fitting an assumed distribution. Having empirical data about a given parameter, one can try to fit these data to one or more of the known statistical distributions. The acceptance or rejection of a specific distribution to the data can be based on statistical tests for "goodness of fit". A list of possible statistical distributions and the estimation of their parameters from sample data can be found in [10]. (3) Pearson system [10, 22]. According to Pearson, a group of distribution families can be generated as a solution to the differential equation: dg(x) _ dx

(x - a ) g ( x ) - b o + b ~ x + b 2 x 2 + " • ""

(19)

Each family of these distributions can be generated by the proper choice of a and b/s, the Pearson Parameters, which are a function of the sample moments. The solution of eqn (19) leads to a large number of distribution families, including the normal, Beta (Pearson Type I), and Gamma distributions (Pearson Type III). The mode of the Pearson distribution is at the point x=a.

Pearson derived thirteen distribution types which can be found with their applications in [10, 22]. 2.3. Investment selection After determining the distribution of the NPV of the different investment opportunities, two major decisions may be involved: (a) To accept or reject the project. (b) If accepted, to rank those projects that are acceptable. In the absence of a utility function, against which all projects can be evaluated, the acceptance or rejection of the project and the ranking of projects by the decision maker will be subjective in nature. However, consistent with a rational decision policy, the acceptance or rejection of a project "A", and the preference of a project "B" over "A" may be decided according to one or more of the non-exhaustive set of rules outlined in the following paragraph. Denoting: f ( x ) = probability density function of x, mi(x) = ith moment of x, E ( x ) = expected value of x,

tr(x) = standard deviation of x, maximum acceptable ratio on x, Rmin(x) =-minimum acceptable ratio on x, V =-Net Present Value, NPV, V~,i. = minimum acceptable NPV. L = Loss, G = Gain, L~ax =- maximum acceptable loss, Gmi. ~ minimum acceptable gain, D -= odds of gain, Dm~.- minimum acceptable odds in favor of investment, U--- utility, Umj, -= minimum acceptable utility.

Rma×(X) ~

Evaluationof risk in investmentalternatives

189

2.4. Reiection o[ an investment Investment "A" may be rejected if one or more of the following conditions occur: (a) Minimum acceptable NPV E(V,,,) < V,.,.

(20)

E(v)=f~=Vf(V)dV.

(21)

where

(b) Maximum tolerable variance ratio o'(VA) . . . . .

E-----~A) "~" -/~max{,V)

(22)

o'2(V) = f~® V:f(V)dV.

(23)

where

(c) Maximum allowable expected loss E(La) > Lmax(V)

(24)

E(L) = J_. IVII(V) d V.

(25)

where

(d) Maximum allowable loss ratio E(LA) > Rmax(L).

(26)

E(GA) < G

(27)

E(G) = fo® V[(V)dV.

(28)

(e) Minimum acceptable gain

where

(f) Minimum allowable gain ratio E(GA) <

(29)

(g) Minimum odds on gain

Da < D,.i.

(30)

D=P(G)

(31)

where

p(L)

190

M. SADEKE1D and H. K. ELDIN

and p(L)

/(V)dV

(32)

p(G) = fo~ f(V) d r .

(33)

UA < U~..

(34)

(h) Minimum acceptable utility

2.5. Ranking of investments Investment B is preferable than A if one or more of the following conditions occur:

(a)

E( V A) < E( VB)

(b)

E(V.~ > EtVB~)

(36)

(c)

E(L,) > E(L,)

(37)

(d)

E ( L , ) > E(LB) ~ EtVB)

(38)

(e)

E( G ^) < E( Ga)

(39)

~(V.)

~(V~)

E(GA) _ E(GB)

E(V~) < ~

(f)

(35)

(40)

(g)

DA < DB

(41)

(h)

U, < UB

(42)

2.6. Utility models When a utility function that represents the investor's attitude is available, different investment opportunities are evaluated against this function. An investment opportunity is accepted if it scores higher than a minimum utility measure decides by the investor. Also, a project "A" is preferred to project "B" if the utility of "A" is higher than that of "B". Three utility are presented here. To discuss these models, let us define the following nomenclature: E(U) = = ml = or = a, b, c, d =

expected utility of the return of a particular portfolio, mean return for the portfolio, ith moment, standard deviation of the return for the portfolio, constants, and U(V) = utility of V. (1) Farrar and Freund. Farrar[8] and Freund[9] used a utility function of the form: U(V) = 1 - e-°'.

(43)

(2) Cramer and Smith. Cramer and Smith[5] used a model of the form: E[U(V)] = lz - atrbI c

(44)

where I is the amount of investment in the project and/x = E(V). (3) Bussey. Bussey[1] developed a quartic utility function as a general model for both risk-averse investors as well as nonrisk avoiding ones. His model includes third and fourth

Evaluation of risk in investment alternatives

191

moments as follows: U(V)= aV- bW + cW- dV 4 E[ U( V)] = a E ( V ) - b E ( W ) + cE( V 3) - dE( V)"

(45)

(46)

where a, b, c, d are constants. Substituting for: E(V) = #

(47)

E ( V ~) = ~=+#2

(48)

E ( V 3) --. 30.2# + # 3 + m3

(49)

E(V4) = #4 + 60r2#2 + 4m3# + m4

(50)

E[ U( V)] = - d # 4 + c# 3- (b + 6d0"2)#2 + (a + 3c0"2- 4din3)# -

(bo"2- cm3 + din4)

(51)

where # = E(V)

(52)

or~ = E ( V - #)2

(53)

m3 = E( V 3)

(54)

m4 = E(W).

(55)

3. UPFAR: UTILITY PROGRAM FOR THE ANALYSIS OF RISK

The UPFAR package consists of four major modules serving the four basic functions in a risk analysis study: 1. Probability distribution fitting: empirical data that are compiled from historical experiences are run through the program PRODIP (Probability D/stribution Program) to get the best fit of a predecided distribution or, to get the best probability distribution that fits the data. 2. Simulation: GUSS (General Utility Simulation System) samples from the different distributions, links with the problem's model, and performs housekeeping operations for inputs and outputs. 3 . Investment model: INMO (Investment Model) includes the different mathematical relationships between the variables of the problem under study. 4 . Utility Evaluation and Measurement: RUM (Routine for Utility Measurement) translates the probability distributions of the output parameters into utility measures. The functional relationships between the UPFAR modules is given in Fig. 1. In general, a risk analysis study goes through the following procedure: 1. Collect data related to the different parameters to the problem.

UPFAR

£mpiricol j . doto _,-dr%

Stochostic, determi nistic ~lva riobles ond controlsi A ~ I Probability ~ L~ '1 distributions

I PRODlP

I~

J - - ~ ' ] r ~ 7 ~ Utility model ~ P V I .....

;

I

J i r.______m IS°mpl"l I L I ] ..... I

Output

I

distributions

I

Fig. 1. Utility Program For Analysis of Risk (UPFAR)--flow chart.

192

M. SADEKE1D and H. K. Eut)ls

2. Run PRODIP on these data to determine the best probability distribution that fits these data. 3. Enter probabilities obtained from PRODIP, together with other probabilities that may be estimated on a subjective basis into GUSS. 4. Run GUSS together with INMO (or any other special subroutine representing the relationships between the investment parameters) for simulation. 5. Enter the results obtained from GUSS, in the form of a frequency distribution of the investment's NPV, into PRODIP to determine the best fit for these output data. 6. Run the distribution parameters of the NPV, as obtained in step 5, in RUM to obtain a measure of the investment's utility. 7. Make final decisions of whether or not to accept the investment on the basis of the above results and other relevant information. 3.1. PRODIP: Probability D/stribution Fitting Program PRODIP is a program that tries to find the best fit from a family of statistical distributions to historical or empirical data. PRODIP offers two options, one for fitting continuous distributions to the data, and the other option fits discrete distributions (see Fig. 2). Option A. This option of PRODIP tries to find the best fit out of the Pearson family of distributions based on the data. Option B. This option of PRODIP tries to fit empirical data to one or more of the following seven continuous distributions, as requested: 1. Exponential 4. Fisher-Tippett I 2. Normal 5. Fisher-Tippett II 3. Lognormal 6. Weibull 7. Beta. Option C. This option of PRODIP tries to fit empirical data to one or more of the following five discrete distributions, as requested: 1. Poisson 3. Binomial 2. Truncated Poisson 4. Negative Binomial 5. Truncated Negative Binomial. While PRODIP offers these options to the users, it should be reminded that the choice of any given distribution to the data should be dependent on the nature of the variable and the physical properties of the distributions. For example, if the variable can only assume discrete values, it would be inappropriate to fit to it to any of the continuous distributions that are offered in Option B, or a Pearson distribution, offered in Option A, which is also continuous. On the other hand, if the variable can be satisfied with any one of the continuous distributions offered in Option B, one would then select the distribution that best fits the data as measured by the ;¢2 or the Kolmogorov-Smirnov tests provided in the program. 3.2. GUSS: GENERAL Utility Simulation System GUSS is a general utility simulation system that uses Monte Carlo technique. GUSS is designed in a way that makes it easy to use as a general purpose simulator for risk analysis studies as well as other simulation problems. GUSS has the following responsibilities: 1. Accepts input data about the problem's constants, deterministic variables, and probability distributions of the stochastic variables as well as sorting and reporting controls. 2. Generates samples from the different probability distributions. 3. Links with INMO to calculate the resultant NPV (or other parameters that are of interest) of the generated sample. Each time the link is made with INMO is a one simulation iteration. 4. Repeats steps 2 and 3 until the total number of desired iterations as given to the program have been completed. 5. Sorts the output results of the simulation and orders them in a descending order, giving the probability distribution of the output NPV measure. 6. Calculates different statistics about the output distribution as well as a distribution summary. GUSS is characterized by its simple linkage with a subroutine representing the model of the investment problem (or any other simulations problem). Figure 3 gives a flow diagram of GUSS.

Evaluationof risk in investmentalternatives

193

Calculate moments ] ~ . . ~ opt ion A (Pearson)

Pearson system

find best "type" to fit data Option

Yes

A

~'es

Normal Option [3 WeiDul

Lognormol

OPB : option B (continuous)

FisherTippet ~FisherTippet TT

OPBn:option B#n

I OPC: option C P°iss°n

~'J } Trunc°tedpoisson

,OiPCi d:i;~ ri~tne)C ~n

Option C Trunc°ted negatvie I 4 binomial

Neg°tvie binomial

I Bioomiol

I

Fig. 2. ProbabilityDistributionFitting Program(PRODIP)--flowchart.

GUSS has the capability of handling simulation problems having the following characteristics: • • • • • • •

Up to Up to Up to Up to Up to Up to Up to in the • Up to • Up to

50 integer constants. 50 deterministic variables. 50 stochastic variables, continuous or discrete. 25 testing and acceptability criteria. 10 counters. 7 output parameters. 10 reports with 40 footnotes each, 20 of these footnotes may have values calculated simulation process. 1000 simulation iterations. 10 output reports on the simulation results.

194

M. SADEK Ell) and

H. K. ELDIN

: [Read l Controls

~ ~

" Constants t " Oeterrnimstic variables ] " Distributions [, Titles J --Sample from

distributions

~

1r. . . . I

I 0 I Store results of t z this lap q-------4 I L__~ 1 ' i OrO+++,++ 1

Calculate

out put] / / /

ff'-'xx

distributionsand~----~ Tape ) their statistics Jx,~ --I-~ o~tput I I distribution ~ optional

Fig. 3. General Utility Simulation System (GUSS)----flow chart.

• Up to 10 sorts per output report. • Up to 5 output devices, disks and/or tapes for output parameters detailed distributions. 3.3. INMO: Investment Model The variations between different investment problems is so great that no one model can encompass all these variations. This was a major factor in designings INMO as an independent module (a subroutine) that links with GUSS through common names for the input and output variables. The INMO model presented here is applicable to a wide range of investment problems that may have any subset or all of the following variables: (l) Capital investments, (a) Working Capital. (b) Land. (c) Equipment and other depreciable investment items. The following is allowed in the model: (a) Investment items can be owned or leased. (b) A time lag can be assumed before the investment item is installed. (c) INMO automatically replaces items that retire during the life of the project with the same items. (d) Three depreciation methods are available: Straight Line, Sum of the Years Digits, and Double Rate Declining Balance. (2) Sales and revenues. The following is allowed for each product: (a) A time lag until the product is introduced to the market. (b) A growth period to maturity volumes (linear growth rate). (c) A price per unit for each product. (d) A direct cost per unit for each product. (3) Overhead expenses. The following is allowed for each expense item: (a) A time lag until the expense is to be incurred. (b) A growth period to maturity of overhead. (4) Tax rate. The calculations of cash flows are made on an after-tax basis. (5) Net present value. The program can calculate and present the distributions of up to seven NPV's calculated for up to seven rate of return assumptions.

Evaluation of risk in investment alternatives

195

Modifications can be made to the program to enable it to handle variations to the above. The degree of ease or difficulty of such changes are dependent on the variance of the characteristics of the new problem to what the present INMO model can handle. For example, INMO can be easily expanded to allow a growth-function of the maturity volumes, etc. Figure 4 gives a flow diagram of INMO. 3.4 RUM: Routine for Utility Measurement This module includes three utility models: 1. The Farrer model; 2. The Cramer's and Smith's model; 3. The Bussey's model. Data about the NPV distribution statistics are entered to RUM for evaluation and making investment decisions. The constants and parameters of the desired utility model are supplied by the user. Figure 5 gives a flow diagram of RUM. r

~

"-"J Guss I I-. . . . . . . . . . . I Samples L

I

I J

I J

I

Set Guss's controls I

i

_1 Evaluate copi,ol

I

costs cash flows offer

depreciation end tax

1

Evaluate revenues offer tax odd to cash flows

~

__

Evaluate expenses after tax add to cash flows

~ t

Calculate NPV~s

Fig. 4, Investment Model (INMO)--flow chart,

No

~

IYes and statistics I

~Yes ..Readparameters] and statistics

Calculate I

Calculate ]

~eod porome~'ersI

l

L

No

uti ty

l

. j

I

o,i,i,y

j

ryes

!

I~= =rained,'4/ I o,',,~s,otisties / /

1

I Calculate L ~t,,,ty

[

II

I I

.

Fig. 5. Routine for Utility Measurement (RUM)--flow chart.

196

M. SADEK EID and H. K. El~DItq 4. USE OF U P F A R

The use of the UPFAR system is a simple task as a result of its modular design, flexibility, and ease of input/output procedures. The major attention in a risk analysis or other simulation problems should be directed to the conceptual aspects of the problem under study while preparing the input, choosing certain options of the model, and to the analysis of the output. Verification that a chosen distribution, the INMO model, and the chosen utility model, do represent the real life problem is essential. Otherwise, the output results, regardless of how good they look become of no value. To use the UPFAR modules, the following procedure is recommended: 1. Identify the different variables and parameters that are pertinent to the problem. 2. Ascertain that the input data are representative of the variable under study and that no extraneous observations or extreme cases are included. 3. Identify the characteristics of the variable under study and ascertain that the variable is not being misrepresented by a chosen distribution. 4. Validate that the resultant distribution that best fits the data agrees with real life situations. 5. Ensure that you are using the appropriate INMO subroutine and that your problem is being represented by that model. 6. Choose the number of simulation iterations that would provide the desired accuracy. 7. Ascertain that the chosen utility model and its parameters represent the investor's attitudes. 8. When presenting the results of the study, make the decision maker aware that: (a) There is no assurance that the outcome of a recommended decision is guaranteed. (b) Risk analysis does not eliminate uncertainty. It rather leads to a clearer understanding of the result of the interaction between the decision elements. 5. C O N C L U S I O N

In general, a risk analysis problem is too complex to lend itself to an analytical model. Simulation, with the availability of high speed computers and low cost of computation, is an effective technique in solving problems where analytical solutions are untenable. The UPFAR system presented here uses Monte Carlo simulation technique and FORTRAN IV. Monte Carlo technique can be easily understood and imposes no restrictions. FORTRAN IV is familiar to most analysts and its compilers are standard software in most computer installations. These were important factors in the choice of the simulation model presented here. Hexibility of the system is also an important factor, especially in our case here where wide variations exist among possible investment alternatives. Modular design is therefore a major feature of UPFAR, the system presented here. Another necessary flexibility that was taken into consideration is the case of interaction among the different modules of the system. REFERENCES 1. L. E. Bussey, Capital budgeting project analysis and selection with complex utility functions. Unpublished Ph.D. Thesis, Oklahoma State University (July, 1970). 2. L. E. Bussey & G. T. Stevens, Jr.. Net present value from complex cash flow streams by simulation. AIIE Transactions 3, 81 (March, 1971). 3. J.R. Canada & H. W. Wadsworth, Methods for quantifying risk in economic analyses of capital projects. J. ind Engng 19, (January, 1%8). 4. H. Chernoff & L. Moses. Elementary Decision Theory. Wiley, New York (1959), 5. R. H. Cramer & B. E. Smith. Decision models for the selection of research projects. The Engineering Economist, 9, l (Winter, 1964). 6. M, S. Eid, A utility program for the analysis of risk (UPFAR). Unpublished Ph.D. Thesis, Oklahoma State University (1974). 7. 3. M. English, A discount function for comparing economic alternatives. J. ing. Engng 16, (March-April, 1965). 8. D. E. Farrer. The Investment Decision Under Uncertainty. Prentice-H~,ll, Englewood Cliffs, New Jersey, (1%2). 9. R. J. Freund, The introduction of risk into a programming model. Econometrica 24, 253 (July, 1956). 10. G. J. Hahn & S. S. Shapiro, Statistical Models in Engineering. Wiley, New York (1967). 11. D. B. Hertz, Risk analysis in capital investments. Harvard Business Review, 95 (Jan,-Feb., 1964). 12. D. B. Hertz, Investment policies that pay off. Harvard Business Review 96 (Jan.-Feb., 1968). 13. S. W, Hess & H. A. Quigley, Analysis of risk in investiments using Monte Carlo technique. Chem, Engng Prog. Syrup. 59, (19). 14. I. R. Hicks & R. G. D. Allen, A reconsideration of the theory of value. Economica 52, New Series I (Feb., 193,t); and ibid, 1, 196 (May, 1934). 15. F. Hillier, The derivation ol probabilistic information for the evaluation of risky investments. Mgmt Sci. 9,443 (April, 1%3).

Evaluation of risk in investment alternatives

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16. F. Hillier, Supplement to the derivations of probabilistic information for the evaluation of risky investments. Mgmt Sci. 11, 485 (January, 1965). 17. F. Hillier & D. V. Helbink, Evaluating risky capital investments. California Mgmt Rev. 71 (Winter, 1%5-66). 18. I. Horowitz, The plant investment decision revisited. J. ind. Engng 17, 1966 (August, 1%6). 19. B. W. Lindgren Statistical Theory. MacMillan, New York (1%8). 20. H. M. Markowitz, Portfolio Selection: EOicient Diversilication o[ Investments. Wiley, New York (1959). 21. A. Marshall, Principles o[ Economics. MacMillan, London (1920). 22. M. Merrington & E. S. Pearson, An approximation to the distribution of non-central t. Biometrica No. 45,484 (1958). 23. I-I. Raiffa, Risk ambiguity and the savage axioms: comment. Q. Jl Econ. (Nov., 1%1). 24. H. Raiffa, Decision Analysis. Addison-Wesley, Reading, Mass. (1968). 25. A. Reisman & A. K. Rap, Discounted Cash Flow Analysis: Stochastic Extension. Monograph Series # 1, Engineering Economy Division, American Institute of Industrial Engineers, Inc., Atlanta Norcross, Georgia (1973). 26. E. Solomon, Measuring a company's cost of capital. J. Business (October, 1955). 27, R. O. Swalm, Utility theory--insights into risk taking. Harvard Bus. Rev. 123 (Nov.-Dec., 1966). 28. M. Tribus, R. Evans & G. Crellin, The use of entropy in hypothesis testing. 10th Nat. Syrup. on Reliability and Quality Control (Jan. 7-9, 1%4). 29. J. Von Neumann & O. Morgenstern. Theory of Games and Economic Behavior, 2nd ed. Princeton University Press, Princeton, New Jersey (1947).