Monte Carlo Simulation in Risk AnalysisAuthor(s): Hercules E. HaralambidesSource: Financial Management, Vol. 20, No. 2 (Summer, 1991), pp. 15-16Published by: Wiley on behalf of the Financial Management Association InternationalStable URL: http://www.jstor.org/stable/3665724 .

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FM LETTERS 15

Monte Carlo Simulation in Risk Analysis

In the traditional, single estimate, ap- proach to investment appraisal, the project evaluator assigns unique values, his best estimates to the outcomes (net cash flows) of a contemplated invest- ment proposal. On the basis of these out- comes, the evaluator calculates the project's NPV, which, as a result, also assumes a unique value. One of the major criticisms of this approach is that it does not supply the analyst with any in- formation about how safe or confident he can be with the result of his appraisal; something that is particularly important in the case of marginal projects.

In the risk analysis approach which is used here, the analyst, because of his uncertainty, considers a range of possible outcomes for each year's cash flow, each one assigned a certain subjective prob- ability of occurrence. These probabilities are determined by the analyst himself according to his prior information and experience. Thus, instead of a single es- timate, a probability distribution of pos- sible outcomes is considered for every year in the project's life.

In this way, the resulting NPV will also be a discrete random variable. In the case of independent outcomes, its mean and standard deviation (the latter provid- ing a measure of the project's risk) are given by:

E(NPV) = I

t C, t=1 (l+i)t 1t

w oh = (1+i2t (1)

where It and at are the mean and stand- ard deviation of the probability distribu- tion of possible outcomes of year t (t = 1,2,... n), i the discount rate and

Co the initial capital outlay. The probability distribution of NPV

can be approximated by a sampling dis- tribution, obtained with the help of com- puter simulation. A simple, three-period project can be used to demonstrate the simulation procedure. Once this is estab- lished, it can be easily generalized for

Exhibit 1. The H-Line

TheH -Line

0.9-

0.8-

0.7-

0.6-

rY co 0.5

0.4-

0.3-

0.2- Risk= -0.09+2.7(Discount Rate) 0.1-

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Discount Rate

multiperiod projects. The investor's cost of capital is assumed to be 10% and the project involves an initial capital outlay of $100. Seven possible outcomes are as- sumed and the assigned probabilities are also the same for each year.

A random number generator (RNG) is subsequently used to sample from the multivariate probability distribution of possible outcomes. For each resulting sample point (triplet of possible out- comes) the project's NPV is calculated. The experiment is repeated a great num- ber of times and a series of NPVs is obtained that can be both negative and positive. These NPVs are subsequently arranged in a grouped frequency dis- tribution whose mean and standard deviation are found to be equal to $23.4 and $24.4, respectively. The population mean and standard deviation, as they can be calculated from the two relationships above, are equal to $22.7 and $25.3, and they both diverge less than 4% from the corresponding sample figures.

The cumulative probability distribu- tion of NPV is even more illustrative and helpful to the analyst. It shows that in the

present case, the probability of obtaining an NPV of at least 40 is 70%, while the probability of getting a negative NPV is 16%. This last result is what the project evaluator is really concerned about; the probability that, just because of his un- certainty, a selected project turns out to be unprofitable (NPV < 0). This prob- ability can be considered as an alternative definition of the project's risk.

It is rather obvious that the above probability is a function of the discount rate used. In the present case, the 16% risk, as defined above, means that ap- proximately 32 out of the 200 NPVs that were finally generated were less than zero. Had a discount rate higher than the assumed 10% been used, the negative NPVs would, on average, be more than 32 and thus, the project's risk would have been higher, too.

The relationship between the investor's cost of capital (discount rate) and the project's risk (defined as the probability of getting a negative NPV) can be seen with the help of the H-Line in Exhibit 1. To derive the H-Line, the above simulation was repeated 100

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16 FINANCIAL MANAGEMENT/SUMMER 1991

times, each time for a different discount rate in the range (5%, 40%). Each time, 200 NPVs were generated and the nega- tive ones were identified, counted and expressed as a percentage of the total. These percentages (risk values) were then plotted to give the jagged line in Exhibit 1. Finally, the above risk was regressed on the corresponding discount rates to give the regression line in Exhibit 1, which is the H-Line.

The H-Line can be profitably used in three ways: (i) For any given cost of capi- tal, the investor can read off from the vertical axis the risk of obtaining a nega- tive NPV. For example, if the investor in the present case had a cost of capital of say, 20%, the probability of getting a negative NPV would be found to be as high as 45%. (ii) Even more importantly, if the investor has an idea of the maxi- mum risk he would be willing to accept

for the particular project, he can read off from the horizontal axis the maximum cost of capital he should have in order to achieve this. For example, if the investor would not be prepared to accept a project with a risk higher than say 15%, he should only initiate the particular project if he can ensure a cost of capital of less than 9%. (iii) Finally, the es- timated slope coefficient is also worth mentioning: The coefficient shows the sensitivity of risk with respect to changes in the cost of capital. In this case, the coefficient was estimated as being equal to 2.7, meaning that, on average, if the cost of capital increases by 1%, the cor- responding risk will increase by ap- proximately 2.7%.

Dr. Hercules E. Haralambides World Maritime University

Malmoe, Sweden

Errata

Readers should note the following corrections to Anjan Thakor's "Game Theory in Finance," as published in the Spring 1991 Issue of Financial Manage- ment:

(i) On page 82, at the top of the second column, first full paragraph, the sentence beginning "Suppose investors" should read as follows: "Suppose inves- tors assign the belief Pr(defector is ugly | D E (0, D*) observed) = 1." Also, at the end of that same paragraph "D > Dg" should read "D >

D*". (ii) On page 85, in the first column,

Equation (34) should read as follows:

aug - .Fg(Dg) > arb - ;Fg(Db) . (34)

The Spring 1991 Issue had already gone to press by the time the above were brought to our attention.

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