169

HBR Classic Risk analysis incapital investment

David B. Hertz

How can husiness execu-tives make the best in-vestment decisions? Isthere a method of riskanalysis to help managersmake wise acquisitions,launch new products,modernize the plant, oravoid overcapacity?"Risk Analysis in CapitalInvestment" takes a lookat questions such as theseand says "yes"—by mea-suring the multitude ofrisks involved in eachsituation. Mathematicalformulas that predict asingle rate of return or"hest estimate" are notenougb. Tbe author's ap-proach emphasizes the na-ture and processing of thedata used and specificcomhinations of variahleslike cash flow, return oninvestment, and risk toesrimate the odds for eachpotential outcome.Managers can examinethe added informarionprovided in this wayto rate more accuratelythe chances of sub-stantial gain in their ven-tures. Tbe article, original-ly presented in 1964,continues to interest HBRreaders; the more than153,000 reprints sold sincethen testify to the im-

portance of this type oftbinking on investmentanalysis. In a retrospectivecommentary, the authordiscusses the now rou-tine use of risk analysisin husiness and govem-ment, emphasizing thattbe method can—andshould—he used in anydecision-requiring situa-tions in our uncertainworld.

When this artiele was firstpublished, Mr. Hertzwas a principal withMcKinsey & Company,Inc., the managementconsulring firm. He iscurrently a senior direc-tor there as well aschairman of the boardof a new magazine.Prime Time. He is theauthor of a follow-uparticle in HBR entitled"Investment Policies thatPay Off" (January-February1968) in addirion to sever-al hooks, including NewPower for Manage-ment: Computer Systemsand Management Science(McGraw-Hill, 19 69} andThe Theory and Practiceof Industrial Research(McGraw-Hill, 1949).

Of all the decisions that business executives mustmake, none is more challenging—and none has re-ceived more attention—than choosing among alter-native capital investnient opportunities. Whatmakes this kind of decision so demanding, of course^is not the prohlem of projecting return on invest-ment under any given set of assumptions. The dif-ficulty is in the assumptions and in their impact.Each assumption involves its own degree—often ahigh degree—of uncertainty; and, taken, together^these combined uncertainties can multiply into atotal uncertainty of critical proportions. This iswhere the element of risk enters, and it is in theevaluation of risk that the executive has been able toget little help from currently available tools andtechniques.

There is a way to help the executive sharpen keycapital investment decisions by providing him orher with a realistic measurement of the risks in-volved. Armed with this gauge, which evaluates therisk at each possible level of return, he or she isthen in a position to measure more knowledgeablyalternative courses of action against eorporate ob-jectives.

Need for new concept

The evaluation of a capital investment project startswith the principle that the productivity of capitalis measured by the rate of return we expect to re-ceive over some future period. A dollar received nextyear is worth less to us than a dollar in hand today.

Harvard Business Review September-October 1979

Expenditures three years hence are less costly thanexpenditures of equal magnitude two years fromnow. For this reason we cannot calculate the rate ofreturn realistically unless we take into account (a)when the sums involved in an investment are spentand (h) when the returns are received.

Comparing altemative investments is thus com-plicated hy the fact that they usually differ not onlyin size but also in the length of time over which ex-penditures will have to he made and benefits re-turned.

These facts of investment life long ago made ap-parent the shortcomings of approaches that simplyaveraged expenditures and benefits, or lumped them,as in the numher-of-years-to-pay-out method. Theseshortcomings stimulated students of decision mak-ing to explore more precise methods for determiningwhether one investment would leave a companybetter off in the long run than would another courseof action.

It is not surprising, then, that much effort has beenapplied to the development of ways to improve ourability to discriminate among investment alterna-tives. The focus of all of these investigations hasbeen to sharpen the definition of the value of capi-tal investments to the company. The controversyand furor that once came out in the business pressover the most appropriate way of calculating thesevalues have largely heen resolved in favor of thediscounted cash flow method as a reasonable meansof measuring the rate of return that can he expectedin the future from an investment made today.

Thus we have methods which are more or lesselaborate mathematical formulas for comparing theoutcomes of various investments and the combina-tions of the variahles that will affect the invest-ments. As these techniques have progressed, themathematics involved has become more and moreprecise, so that we can now calculate discounted re-turns to a fraction of a percent.

But the sophisticated executive knows that hehindthese precise calculations are data which are notthat precise. At hest, the rate-of-return informationhe is provided with is hased on an average of differ-ent opinions with varying reliabilities and differentranges of probability. When the expected returnson two investments are close, he is likely to he in-fluenced hy intangibles—a precarious pursuit at best.Even when the figures for two investments are quitefar apart, and the choice seems clear, there lurkmemories of the Edsel and other ill-fated ventures.

In short, the decision maker realizes that there issomething more he ought to know, something inaddition to the expected rate of return. What is miss-

ing has to do with the nature of the data on whichthe expected rate of return is calculated and withthe way those data are processed. It involves uncer-tainty, with possihilities and probabilities extendingacross a wide range of rewards and risks. (For asummary of the new approach, see the ruled insert.)

The Achilles heel

The fatal weakness of past approaches thus hasnothing to do with the mathematics of rate-of-retumcalculation. We have pushed along this path so farthat the precision of our calculation is, if anything,somewhat illusory. The fact is that, no matter whatmathematics is used, each of the variahles enteringinto the calculation of rate of return is suhject to ahigh level of uncertainty.

For example, the useful life of a new piece of capi-tal equipment is rarely known in advance with anydegree of certainty. It may he affected hy variationsin ohsolescence or deterioration, and relatively smallchanges in use life can lead to large changes in re-turn. Yet an expected value for the life of the equip-ment—hased on a great deal of data from which asingle hest possible forecast has heen developed—is entered into the rate-of-return calculation. Thesame is done for the other factors that have a sig-nificant hearing on the decision at hand.

Let us look at how this works out in a simple case—one in which the odds appear to be all in favor ofa particular decision. The executives of a food com-pany must decide whether to launch a new packagedcereal. They have come to the conclusion that flvefactors are the determining variahles: advertisingand promotion expense, total cereal market, share ofmarket for this product, operating costs, and newcapital investment.

On the basis of the "most likely" estimate foreach of these variahles, the picture looks very hright—a healthy 30% return. This future, however, de-pends on whether each of these estimates actuallycomes true. If each of these educated guesses has,for example, a 60% chance of heing correct, there isonly an 8% chance that all flve will be correct (.60 X.60 X .60 X .60 X .6o|. So the "expected" return ac-tually depends on a rather unlikely coincidence. Thedecision maker needs to know a great deal moreahout the other values used to make each of the fiveestimates and about what he stands to gain or losefrom various combinations of these values.

This simple example illustrates that the rate ofreturn actually depends on a speciflc combinationof values of a great many different variahles. But

Bummary of new approach

Capital investment

^ate of returnnvestment X

/20 chanceto%

2/10 chance atexpected 9.2%

1/50 chanceat 30%

=1ate of returnnvestment Y

1/10 chanceitO%

2/10 chance atexpected 10.3%

1/10C chanceat 30%

After examining presentmethods of comparing alterna-tive investments, the authorreports on his firm's experiencein applying a new approachto the problem. Using thisapproach, management takesthe various levels of possiblecash flows, return on invest-ment, and other results of aproposed outlay and gets anestimate of the odds for eachpotential outoome.

Currently, many facilities deci-sions are based on discountedcash flow calculations. Manage-ment is fold, for example, thatInvestment X has an expectedinternal rate of return oi 9,2%,while for Investment Y a 10.3%return can be expected.

By contrast, the new approachwould put in front of the execu-tive a schedule that gives himthe mosf likely return from X,but also tells him that X has1 chance in 20 of being a fofalloss, 1 in lOof earning from 4%to 5%, 2 in 10 ot paying from 8%to 10%, and 1 chance in 50 ofattaining a 30% rate of return.

From another schedule helearns what the most likely rateof return is from Y, but also thatY has 1 chance in 10 of resultingin a total loss. 1 in 10 of earningfrom 3% fo 5% return, 2 in 10of paying between 9% and 11%,and 1 chance in 100 of a 30%rate of return.

In this instance, the estimatesof the rates of return provided bythe two approaches would notbe substantially different. How-ever, to the decision maker withfhe added information. Invest-ment Y no longer looks like fheclearly better choice, since withX the chances of substantialgain are higher and the risks ofloss lower.

Two things have made thisapproach appealing to man-agers who have used it:

1. Certainly in every case it isa more descriptive statement ofthe two opportunities. And insome cases it might well reversethe decision, in line with particu-lar corporate objectives,

2. This is not a difficult tech-niqueto use, since much of theinformation needed is alreadyavailable - or readily accessible -and the validity of the principlesinvolved has, for the most part,already been proved in otherapplications.

The enthusiasm with whichmanagements exposed to thisapproach have received it sug-gests that it may have wideapplication. It has particular rele-vance, for example, in suchknotty problems as investmentsrelating fo acquisitions or newproducts and in decisions thatmight involve excess capacity.

only the expected levels of ranges (worst, average,best; or pessimistic, most likely, optimistic) of thesevariables arc used in formal mathematical ways toprovide the figures given to management. Thus pre-dicting a single most likely rate of return gives pre-cise numbers that do not tell the whole story.

The expected rate of return represents only a fewpoints on a continuous curve of possible combina-tions of future happenings. It is a bit like trying topredict the outcome in a dice game by saying that the

most likely outcome is a 7. The description is in-complete because it does not tell us about all theother things that could happen. In Exhibit I, forinstance, we see the odds on throws of only twodice having 6 sides. Now suppose that each of eightdice has 100 sides. This is a situation more compar-able to business investment, where the company'smarket share might become any r of 100 differentsizes and where there are eight factors (pricing,promotion, and so on) that can affect the outcome.

Harvard Business Review September-October 1979

Exhibit IDescribing uncertainty-a throw of the dice

Nor is this the only trouhle. Our willingness tobet on a roll of the dice depends not only oa theodds hut also on the stakes. Since the probahility ofrolling a 7 is I in 6, we might he quite willing torisk a few dollars on that outcome at suitahle odds.But would we be equally willing to wager $ro,oooor $roo,ooo at those same odds, or even at hetterodds? In short, risk is influenced hoth hy the oddson various events occurring and hy the magnitudeof the rewards or penalties that are involved whenthey do occur.

To illustrate again, suppose that a company isconsidering an investment of $i million. The bestestimate of the probable retum is $200,000 a year.It could well he that this estimate is the average ofthree possible returns—a i-in-3 chance of getting noreturn at all, a r-in-3 chance of getting $200,000per year, a i-in-3 chance of getting $400,000 per year.Suppose that getting no retum at all would put thecompany out of husiness. Then, by accepting thisproposal, management is taking a i-in-3 ehanee ofgoing hankrupt.

If only the hest-estimate analysis is used, how-ever, management might go ahead, unaware that itis taking a big chance. If all of the availahle infor-mation were examined, management might preferan altemative proposal with a smaller, but morecertain (that is, less variable) expectation.

Such considerations have led almost all advocatesof the use of modem capital-investment-index cal-culations to plead for a recognition of the elements

penetrable mists of any forecast."How can exeeutives penetrate the mists of uncer-

tainty surrounding the choices among alternatives?

Limited improvements

A numher of efforts to cope with uncertainty haveheen successful up to a point, hut all seem to fallshort of the mark in one way or another:

1. More accurate forecasts—Reducing the error inestimates is a worthy ohjective. But no matter howmany estimates of the future go into a capital invest-ment decision, when all is said and done, the futureis still the future. Therefore, however well we fore-cast, we are still left with the certain knowledgethat we eannot eliminate all uncertainty.

2. Empirical adjustments—Adjusting the factors in-fluencing the outcome of a deeision is suhjeet toserious difficulties. We would like to adjust them soas to cut down the likelihood that we will makea "had" investment, hut how can we do that with-out at the same time spoiling our chances to make a"good" one? And in any case, what is the basis for

1. "The fndgment FactorP- 99-2. "Monitoring Capital Ii

t Decisions," HBR March-Apri! 1961,

tments," Financial Excf

Capital Budgeting and Game Theory," HBR Noi

m, April i96i, p. i9-

ber-December 1956,

Capital investment

adjustment? We adjust, not for uncertainty, butfor bias.

For example, construction estimates are often ex-ceeded. If a company's history of construction costsis that 90% of its estimates have been exceeded by15%, then in a capital estimate there is every jus-tification for increasing the value of this factor by15%. This is a matter of improving the accuracy ofthe estimate.

But suppose that new-product sales estimates havebeen exceeded by more than 75% in one-fourth ofall historical cases and have not reached 50̂ 0 of theestimate in one-sixth of all such cases? Penalties forsuch overestimating are very real, and so manage-ment is apt to reduce the sales estimate to "cover"the one case in six—thereby reducing the calculatedrate of return. In so doing, it is possibly missingsome of its best opportunities.

3. Revising cutoff rates—Selecting higher cutoffrates for protecting against uncertainty is attempt-ing much the same thing. Management would liketo have a possibility of retum in proportion to therisk it takes. Where there is much uncertainty in-volved in the various estimates of sales, costs, prices,and so on, a high calculated retum from the invest-ment provides some incentive for taking the risk.This is, in fact, a perfectly sound position. Thetrouble is that the decision maker still needs toknow explicitly what risks he is taking—and whatthe odds are on achieving the expected return.

4. Thiee-level estimates—A start at spelling outrisks is sometimes made by taking the high, me-dium, and low values of the estimated factors andcalculating rates of return based on various combi-nations of the pessimistic, average, and optimisticestimates. These calculations give a picture of therange of possible results but do not tell the execu-tive whether the pessimistic result is more likelythan the optimistic one—or, in fact, whether theaverage result is much more likely to occur thaneither of the extremes. So, although this is a step inthe right direction, it still does not give a clearenough picture for comparing alternatives.

5. Selected probabilities—Various methods havebeen used to include the probabilities of specific fac-tors in the retum calculation. L. C. Grant discusseda program for forecasting discounted cash flow ratesof retum where the service life is subject to obsoles-cence and deterioration. He calculated the odds thatthe investment will terminate at any time after it ismade depending on the probability distribution ofthe service-life factor. After having calculated thesefactors for each year through maximum service life,he determined an overall expected rate of return.^

Edward G. Bennion suggested the use of gametheory to take into account altemative marketgrowth rates as they would determine rate of retumfor various options. He used the estimated probabil-ities that specific growth rates would occur to de-velop optimum strategies. Bennion pointed out:

"Forecasting can result in a negative contributionto capital budget decisions unless it goes furtherthan merely providing a single most probable pre-diction. . . . [with] an estimated probability coeffi-cient for the forecast, plus knowledge of the payoffsfor the company's alternative investments and cal-culation of indifference probabilities... the marginof error may be substantially reduced, and the busi-nessman can tell just how far off his forecast maybe before it leads him to a wrong decision." '

Note that both of these methods yield an expectedreturn, each based on only one uncertain input fac-tor—service life in the first case, market growth inthe second. Both are helpful, and both tend to im-prove the clarity with which the executive can viewinvestment altematives. But neither sharpens up therange of "risk taken" or "retum hoped for" suffi-ciently to help very much in the complex decisionsof capital planning.

Sharpening the picture

Since every one of the many factors that enter intothe evaluation of a decision is subject to some un-certainty, the executive needs a helpful portrayal ofthe effects that the uncertainty surrounding each ofthe significant factors has on the returns he is like-ly to achieve. Therefore, I use a method combiningthe variabilities inherent in all the relevant factorsunder consideration. The objective is to give a clearpicture of the relative risk and the probable odds ofcoming out ahead or behind in light of uncertainforeknowledge.

A simulation of the way these factors may com-bine as the future unfolds is the key to extractingthe maximum information from the available fore-casts. In fact, the approach is very simple, using acomputer to do the necessary arithmetic. To carryout the analysis, a company must follow three steps:

1. Estimate the range of values for each of the fac-tors (for example, range of selling price and salesgrowth rate) and within that range the likelihoodof occurrence of each value.

Harvard Business Review September-October 1979

2. Select at random one value from the distributionof values for each factor. Then combine the valuesfor all of the factors and compute the rate of retum(or present value] from that combination. For in-stance, the lowest in the range of prices might becombined with the highest in the range of growthrate and other factors. (The fact that the elementsare dependent should be taken into account, as weshall see later.]

3. Do this over and over again to define and eval-uate the odds of the occurrence of each possiblerate of return. Since there are literally millions ofpossible combinations of values, we need to testthe likelihood that various returns on the invest-ment will occur. This is like finding out by record-ing the results of a great many throws what percentof 7S or other combinations we may expect in toss-ing dice. The result will be a listing of the rates ofretum we might achieve, ranging from a loss (if thefactors go against us] to whatever maximum gain ispossible with the estimates that have been made.

For each of these rates we can determine thechances that it may occur. (Note that a specific re-turn can usually be achieved through more than onecombination of events. The more combinations fora given rate, the higher the chances of achieving it—as with 7s in tossing dice.) The average expectationis the average of the values of all outcomes weightedby the chances of each occurring.

We can also determine the variability of outcomevalues from the average. This is important since, allother factors being equal, management would pre-sumably prefer lower variability for the same re-tum if given the choice. This concept has alreadybeen applied to investment portfolios.

When the expected return and variability of eachof a series of investments have been determined,the same techniques may be used to examine theeffectiveness of various combinations of them inmeeting management objectives.

Practical test

To see how this new approach works in practice, letus take the experience of a management that hasalready analyzed a specific investment proposal byconventional techniques. Taking the same invest-ment schedule and the same expected values ac-

tually used, we can find what results the new meth-od would produce and compare them with the re-sults obtained by conventional methods. As we shallsee, the new picture of risks and returns is differentfrom the old one. Yet the differences are attributablein no way to changes in the basic data—only to theincreased sensitivity of the method to management'suncertainties about the key factors.

Investment proposal

In this case, a medium-size industrial chemical pro-ducer is considering a $ro million extension to itsprocessing plant. The estimated service life of thefacility is ten years; the engineers expect to use150,000 tons of processed material worth $510 perton at an average processing cost of $435 per ton. Isthis investment a good bet? In fact, what is the re-turn that the company may expect? What are therisks? We need to make the best and fullest use ofall the market research and financial analyses thathave been developed, so as to give management aclear picture of this project in an uncertain world.

The key input factors management has decidedto use are market size, selling prices, market growthrate, share of market (which results in physical salesvolume), investment required, residual value of in-vestment, operating costs, fixed costs, and usefullife of facilities. These factors are typical of those inmany company projects that must be analyzed andcombined to obtain a measure of the attractivenessof a proposed capital facilities investment.

Obtaining estimates

How do we make the recommended type of analysisof this proposal? Our aim is to develop for each ofthe nine factors listed a frequency distribution orprobability curve. The information we need in-cludes the possible range of values for each factor,the average, and some idea as to the likelihood thatthe various possible values will be reached.

It has been my experience that for major capitalproposals managements usually make a significantinvestment in time and funds to pinpoint informa-tion about each of the relevant factors. An objectiveanalysis of the values to be assigned to each can,with little additional effort, yield a subjective proba-bility distribution.

Specifically, it is necessary to probe and questioneach of the experts involved—to find out, for ex-ample, whether the estimated cost of production

Capital investment 175

really can be said to be exactly a certain value orwhether, as is more likely, it should be estimated tolie within a certain range of values. Managementusually ignores that range in its analysis. The rangeis relatively easy to determine; if a guess has to bemade—as it often does—it is easier to guess with someaccuracy a range rather than one specific value. Ihave found from experience that a series of meet-ings with management personnel to discuss suchdistributions are most helpful in getting at realisticanswers to the a priori questions. (The term realisticanswers implies all the information managementdoes not have as well as all that it does have.)

The ranges are directly related to the degree ofconfidence that the estimator has in the estimate.Thus certain estimates may be known to be quiteaccurate. They would be represented by probabilitydistributions stating, for instance, that there is onlyr chance in io that the actual value will be differ-ent from the best estimate by more than io%. Othersmay have as much as ioo% ranges above and belowthe best estimate.

Thus we treat the factor of selling price for thefinished product by asking executives who are re-sponsible for the original estimates these questions;

> Given that $510 is the expected sales price,what is the probability that the price will exceed$550?

> Is there any chance that the price will exceed$650?

> How likely is it that the price will drop below$475?

Managements must ask similar questions for all ofthe other factors until they can construct a curvefor each. Experience shows that this is not as dif-ficult as it sounds. Often information on the degreeof variation in factors is easy to obtain. For instance,historical information on variations in the price ofa commodity is readily available. Similarly, manage-ments can estimate the variability of sales from in-dustry sales records. Even for factors that have nohistory, such as operating costs for a new product,those who make the average estimates must havesome idea of the degree of confidence they have intheir predictions, and therefore they are usually onlytoo glad to express their feelings. Likewise, the lessconfidence they have in their estimates, the greaterwill be the range of possible values that the variablewill assume.

This last point is likely to trouble businessmen.Does it really make sense to seek estimates of varia-tions? It cannot be emphasized too strongly that theless certainty there is in an average estimate, the

Exhibit IISimulation for investment planning

Chances that valuewill be achieved(vertical axis)Range of values(horizontal axis)

,

Market size

• '

Share of market

•

Operating costs

Select-at ^ ^random - sets o ^ ^ | ^these factors ^ ^ Baccording to the ^^Mchances they ^ Hhave of turning ^ ^ Bup in the future ^^M

Probability ^ kvalues for ^ Bsignificant ^ Hfactors ^ H

i

Selling prices

_ ,

•••

Investmentrequired

Fixed costs

3 ^Determine rate ^ Lof return for e a c h ^ |combination ^M

Chances that rate will beachieved (vertical axis)

/

•

*

Rate of return (horizontal axis)

1'

Market growth rate

. • ' • ' * .

/

Residual value ofinvestment

Useful life offacilities

4 w^Repeat process ^ ^

k to give a clear ^ ^1 portrayal of ^ H• — ^ investment risk ^ H

' ' \

%^

\^

\

Harvard Business Review September-Oetober 1979

more important it is to consider the possihle varia-tion in that estimate.

Further, an estimate of the variation possihle ina factor, no matter how judgmental it may be, isalways better than a simple average estimate, sinceit includes more information about what is knownand what is not known. This very lack of knowledgemay distinguish one investment possibility from an-other, so that for rational decision making it mustbe taken into account.

This lack of knowledge is in itself important in-formation about the proposed investment. To throwany information away simply because it is highlyuncertain is a serious error in analysis that the newapproach is designed to correct.

3. Operating and fixed costs—These also are sub-ject to uncertainty but are perhaps the easiest toestimate.

These categories are not independent, and for re-alistic results my approach allows the various fac-tors to be tied together. Thus if price determines thetotal market, we first select from a probability dis-tribution the price for the specific computer runand then use for the total market a probabihty dis-tribution that is logically related to the price se-lected.

We are now ready to compare the values obtainedunder the new approach with those obtained by theold. This comparison is shovm in Exhibit III.

Computer runs

The next step in the proposed approach is to deter-mine the returns that will result from random com-binations of the factors involved. This requires re-alistic restrictions, such as not allowing the totalmarket to vary more than some reasonable amountfrom year to year. Of course, any suitable methodof rating the retum may be used at this point. In theactual case, management preferred discounted cashflow for the reasons cited earlier, so that method isfollowed here.

A computer can be used to carry out the trialsfor the simulation method in very little time andat very little expense. Thus for one trial 3,600 dis-counted cash flow calculations, each based on a se-lection of the nine input factors, were run in twominutes at a cost of $r5 for computer time. The re-sulting rate-of-retum probabilities were read outimmediately and graphed. The process is shownschematically in Exhibit II.

Data compaiisons

The nine input factors described earlier fall intothree categories:

1. Market analyses—Included are market size,market growth rate, the company's share of themarket, and selling prices. For a given combinationof these factors sales revenue may be determined fora particular business.

2. Investment cost analyses-Being tied to thekinds of service-life and operating-cost character-istics expected, these are subject to various kinds oferror and uncertainty^ for instance, automationprogress makes service life uncertain.

Valuable results

How do the results under the new and old ap-proaches compare? In this case, management hadbeen informed, on the basis of the one-best-esti-mate approach, that the expected return was25.3% before taxes. When we run the new set ofdata through the computer program, however, weget an expected retum of only r4.6% before taxes.This surprising difference results not only from therange of values under the new approach but alsofrom the weighing of each value in the range bythe chances of its occurrence.

Our new analysis thus may help management toavoid an unwise investment. In fact, the general re-sult of carefully weighing the information and lackof information in the manner I have suggested is toindicate the true nature of seemingly satisfactory in-vestment proposals. If this practice were followed,managements might avoid mueh overcapacity.

The computer program developed to carry out thesimulation allows for easy insertion of new vari-ables. But most programs do not allow for de-pendence relationships among the various inputfactors. Further, the program used here permits thechoice of a value for price from one distribution,which value determines a particular probability dis-tribution [from among several) that will be used todetermine the values for sales volume. The followingscenario shows how this important techniqueworks:

Suppose we have a wheel, as in roulette, with thenumbers from o to r 5 representing one price for theproduct or material, the numbers 16 to 30 represent-ing a second price, the numbers 3r to 45 a thirdprice, and so on. For each of these segments wewould have a different range of expected market

Capital investment

volumes—for example, $r5o,ooo-$2,oo,ooo for thefirst, $ioo,ooo-$i50,ooo for the second, $75,000-$100,000 for the third. Now suppose we spin thewheel and the ball falls in 37. This means that wepick a sales volume in the $75,ooo-$ioo,ooo range.If the ball goes in rr, we have a different price, andwe turn to the $r50,ooo-$200,ooo range for a salesvolume.

Most significant, perhaps, is the fact that the pro-gram allows management to ascertain the sensitivityof the results to each or all of the input factors.Simply by running the program with changes inthe distribution of an input factor, it is possible todetermine the effect of added or changed information(or lack of information!. It may tum out that fairlylarge changes in some factors do not significantlyaffect the outcomes. In this case, as a matter of fact,management was particularly concerned about thedifficulty in estimating market growth. Running theprogram with variations in this factor quickly dem-onstrated that for average annual growth rates fromi% to s% there was no significant difference in theexpected outcome.

In addition, let us see what the implications areof the detailed knowledge the simulation methodgives us. Under the method using single expectedvalues, management arrives only at a hoped-for ex-pectation of 23.2% after taxes (which, as we haveseen, is wrong unless there is no variability in themany input factors—a highly unlikely event).

With the proposed method, however, the uncer-tainties are clearly portrayed, as shown in ExhibitIV. Note the contrast with the profile obtained underthe conventional approach. This concept has beenused also for evaluation of product introductions,acquisition of businesses, and plant modernization.

Comparing opportunities

From a decision-making point of view one of themost significant advantages of the new method ofdetermining rate of return is that it allows manage-ment to discriminate among measures of (1| ex-pected return based on weighted probabilities of allpossible returns, (2) variability of retum, and (3|risks.

To visualize this advantage, let us take an examplebased on another actual case but simplified for pur-poses of explanation. The example involves two in-

Exhibit IIIComparison of expected values under old and new approaches

Market analyses

1. Market size

Expected value (in tons)

Range

2. Selling prices

Expected value (in dollars/ton)

Range

3. Market growth rate

Expected value

Range

4. Eventual shareof market

Expected value

Range

Investment cost analyses

5. Total investment required

Expected value (in $ millions)

Range

6. Useful life of facilities

Expected value (in years)

Range

7. Residual value (at 10 years)

Expected value (in $ millions)

Range

Other costs

8. Operating costs

Expected vaiue (in dollars/ton)

Range

9. Fixed costs

Expected value (in $ thousands)

Range

Note: Range figures in righl-hand column rties. That is, there is oniy a l-in-100 chancerespectively greater or less than the range.

Conventionalbest estimate"approach

250,000

-

$510

-

3%

-

12%

-

$9.5

10

-

$4.5

-

$435

$300

-

epresent approximately t

New approach

250,000

100,000-340,000

$510

$385-$575

3%

0-6%

12%

3%-17%

$9,5

$7,0-$10,5

10

5-15

$4.5

$3',5-$5.0

$435

$370-$545

$300

$250-$375

%to99%probabili-

vestments under consideration, A and B. With theinvestment analysis, we obtain the tabulated andplotted data in Exhibit V. We see that:

D Investment B has a higher expected retum thanInvestment A.

D Investment B also has substantially more vari-ability than Investment A. There is a good chancethat Investment B will earn a return quite different

178 Harvard Business Review September-October 1979

Exhibit IVAnticipated rates of retum under old and new approaches

Chances that rate of return will be achieved or hettered Concluding note

Conventional analysiswith expected values only

New analysissimulating all Input factors

-10%Anticip

Percent return

0%

5

10

15

20

25

30

-5 0ated rate of returr

51

10 15 20 25 30

Probability of achievingat least the returnshown

96.5%

S0.6

75.2

53,8

43.0

12.6

0

from the expected return of 6.8%—possibly as highas 15% or as low as a loss of 5%. Investment A isnot likely to vary greatly from the anticipated 5%retum.

• Investment B involves far more risk than doesInvestment A. There is virtually no chance of in-curring a loss on Investment A. However, there is ichance in 10 of losing money on Investment B. Ifsuch a loss occurs, its expected size is approximately$200,000.

Clearly, the new method of evaluating investmentsprovides management with far more informationon which to hase a decision. Investment decisionsmade only on the basis of maximum expected re-turn are not unequivocally the best decisions.

The question management faces in selecting capitalinvestments is first and foremost: What informationis needed to clarify the key differences amongvarious altematives? There is agreement as to thebasic factors that should be considered—markets,prices, costs, and so on. And the way the future re-tum on the investment should be calculated, if notagreed on, is at least limited to a few methods, anyof which can be consistently used in a given com-pany. If the input variables turn out as estimated,any of the methods customarily used to rate invest-ments should provide satisfactory [if not necessarilymaximum! returns.

In actual practice, however, the conventionalmethods do not work out satisfactorily. Why? Thereason, as we have seen earlier in this article andas every exeeutive and economist knows, is thatthe estimates used in making the advance calcula-tions are fust that—estimates. More accurate esti-mates would be helpful, but at best the residual un-certainty can easily make a mockery of corporatehopes. Nevertheless, there is a solution. To collectrealistic estimates for the key factors means to findout a great deal about them. Hence the kind of un-certainty that is involved in each estimate can beevaluated ahead of time. Using this knowledge ofuncertainty, executives can maximize the value ofthe information for decision making.

The value of computer programs in developingclear portrayals of the uncertainty and risk sur-rounding altemative investments has been proved.Such programs can produce valuable informationabout the sensitivity of the possible outcomes to thevariability of input factors and to the likelihood ofachieving various possible rates of retum. This in-formation can be extremely important as a backupto management judgment. To have calculations ofthe odds on all possible outcomes lends some assur-ance to the decision makers that the available in-formation has been used with maximum efficiency.

This simulation approach has the inherent ad-vantage of simplicity. It requires only an extensionof the input estimates (to the hest of our ability) interms of probabilities. No projeetion should be piii-pointed unless we are certain of it.

The discipline of thinking through the uncertain-ties of the problem will in itself help to ensure im-provement in making investment choices. For tounderstand uncertainty and risk is to understand

Capital investment

the key business problem—and the key business op-portunity. Sinee the new appioaeh ean be appliedon a continuing basis to each capital alternative asit comes up for consideration and progresses towardfruition, gradual progress may be expected in im-proving the estimation of the probabilities of vari-ation.

Lastly, the eourage to act boldly in the face ofapparent uncertainty can be greatly bolstered by theclarity of portrayal of the risks and possible rewards.To achieve these lasting results requires only aslight effort beyond what most companies alreadyexert in studying capital investments.

Retrospective commentary

when this artiele was published 15 years ago, therewere two recurrent themes in the responses of themanagement eommunity to it: (1| how the uncer-tainties surrounding each key element of an invest-ment decision were to be determined, and |2) whateriteria were to be used to deeide to proceed withan investment once the uncertainties were quanti-fied and displayed.

I answered the latter question in an HBR sequel,"Investment Policies Tbat Pay Off," deseribing therelationships of risks and stakes to longer term in-vestment eriteria. This artiele, published in 1968,showed bow risk analyses can provide bases for de-veloping policies to cboose among a variety of in-vestment alternatives. Similar approaehes were sub-sequently developed for investment fund portfoliomanagement.

The analysis of uneertainty in deseribing complexdecision-making situations is now an integral partof business and government. The elements of aninvestment decision—private or publie—are subjeetto all the uncertainties of an unknown future. Asthe 1964. artiele showed, an estimated probabilitydistribution paints tbe clearest picture of all pos-sible outeomes. Sueh a description contains con-siderably more information tban simplistic com-binations of subjective best estimates of input fac-tors. Best estimates are point estimates [tbere maybe more than one—high, medium, low| of the valueof an element of the investment analysis used fordetermining an outcome decision criterion, such asinternal rate of return or present value of the in-vestment.

Exhibit VComparison of two investment opportunities

Selected statistics

Amount of investment

Life of investment {in years)

Expected annual net cash inflow

Investment A

$10,000,000

10

$ 1,300,000

Investment B

$10,000,000

10

$ 1,400,000

Variability of cash inflow

1 chance in50 of being greaferfhan

1 chance in50 of being/ess fhan*

$ 1,700,000

$ 900,000

$ 3,400,000

($ 600,000)

Expected return on investment

Variability ol return on investment

1 chance in 50 of being greater than

1 chance in 50 of being/ess fftan'

Risk of investment

Chances of a loss

Expected size of toss

' In the case of negative figures (indicaledby (

5.0%

7.0%

3.0%

Negligible

Negligible

jarentheses) less than m

Chances that rate of return will be achieved or bettered

15,5%

(4.0%)

1 inio

$200,000

leans worse ttian-

:'•'- y'O

60 7

40

20

0

\

• " , Investment B

-10% -5 0 5 10 15 20Percent of return on investment

Thus even where the conventional approaeh wasused for tbe best estimate in a single-point determi-nation for tbe statistieally estimated expeeted valuesfrom a distribution of an element, tbe single-pointapproacb was sbown to be exeeedingly misleading.In Exhibit III, a single-point best-estimate analysisgave an internal rate of return of 25.1%. And arisk analysis employing estimated frequency dis-tributions of the elements showed that an averageof possible outcomes, weighted by the relative fre-quency of tbeir oeeurrences at r4.6%, was more re-alistic as well as significantly diflerent. It presenteda truer pieture of the actual average expectation of

Harvard Business Review September-October 1979

the result of this investment (if it eould be repeatedover and over again).

The case was thus made, and the point of thisresult—that risk and uncertainty were more ac-curately defined hy a simulation of input variables-was little questioned thereafter. Managements beganto adopt some form of this procedure to examinesome, if not all, significant investments wheredoubt existed about the risk levels involved. My se-quel article attempted to demonstrate that if enoughinvestments were chosen consistently on the basisof criteria related to these kinds of risk portrayals,the overall outcomes would stabilize around the de-sired expected value or best estimate of the criterion.

All this now seems simple and straightforward.Earlier it was falsely thought that risk analysiswas aimed at eliminating uncertainty, which was notworth doing at all since the future is so desperatelyuncertain. Thus in 1970 the Financial Times [ofLondon) published an article intended to show thefutility of risk analysis. It concemed a baker of geri-atric biscuits who made an investment only to gobankrupt when his nursing home market precipi-tately disappeared with the death of its founder.The author cited as a moral, "Don't put all yourdough in one biscuit."

It took a while for the points to diffuse throughexecutive circles that (1) exactly such an analysiswould have been just as bad, or worse, done viasingle-point subjective estimates, and (2) no oneanalytical technique could control future events,even with sensitive inputs and requirements forfollow-up control to improve the odds as projectedhy the original risk analyses. But in the end, judg-ment would be required in both input estimationand decision.

I did not intend the article to be an argument inmethodology but rather a cautionary note to ex-amine the data surrounding an investment proposalin light of all the pervasive uncertainties in theworld, of which business is simply one part. Theyears since 1964 have made it clear to me that thismessage should have been amplified and more em-phatically insisted on in the article.

Had this point been clearer, the issue whetherto take the risk and proceed with an investmentmight have been less troublesome. Had I been ableto look with more prescience, I might have seen thatthe area of risk analysis would become routine inbusiness and virtually universally adopted in publiccost-benefit issues.

Cost-benefit analysis for public decisions is, ofcourse, only a special form of investment analysis.Goverrmient issues that require decisions involv-

ing significant uncertainty are too numerous tocatalog fuUy-energy, from both fossil and nuclearsources; chemical, drug, and food carcinogen haz-ards; DNA manipulation and its progeny of genesplicing.

The Three Mile Island nuclear accident broughthome the fallibility of stating a risk analysis con-clusion in simplistic terms. The well-known Ras-mussen report on nuclear reactor safety, commis-sioned by the Nuclear Regulatory Commission, un-dertook what amounted to a risk analysis that wasintended to provide a basis for investment decisionsrelating to future nuclear energy production. TheNuclear Regulatory Commission, in January 1979,disclaimed the risk estimates of that report; newstudies to estimate risk are now underway. Butthere is also a school of thought saying we face toomany risks each day to worry about one more.

A commonly stated estimate of the risk of a majornuclear power plant accident is r chance in r,000,000years. In the r964 article, I portrayed the image ofrisk with a chart of the throws of two dice thatwould be required to give various outcomes-.-.fromtwo IS to two 6s, each of these having a r-in-36chance of occurring. There should be no problemin visualizing or testing the meaning and thechances of any of the events pictured hy these dice.And, although r in r,000,000 is somehow presentedas "mind boggling" compared with r in 36, and sounlikely to occur as to be beyond our ken, I sug-gest that it is just as simply visualized.

We simply need to use eight dice at once. If wechart all the possible outcomes for eight dice, as wedid for the two, we find that the sum of 8 (or 48)can occur just one way—via all rs (or all 6s). Theodds of this occurring are roughly r in r,680,000.Thus the visualization of such odds, and moreimportant, the lesson we must learn ahout risk—which incidents like Three Mile Island should teachus—is that what can happen will happen if we justkeep at it long enough. Any of us can simulate astatistical picture of the estimated risks or even thecomplexities of the Rasmussen analysis with enoughpatience and enough dice (or a computer).

Incidentally, to make the eigjit dice act more likethe odds of r in r,000,000, simply mark any two"non-r" sides with a felt pen and count them as rsif they turn up; the odds of getting all rs become alittle less than r in i,roo,ooo. And the chances ofhuman error can he included by similarly markingother dice in the set. The difficulty is not in con-structing such a simulation to portray the odds butin determining events that may lead to these oddsand estimating the frequencies of their occurrence.

Capital investment

Risk analysis has become one with public poUcy.Without it, any important choice that leads to un-certain outcomes is uninformed; with it, properlyapplied and understood, the decision maker—busi-ness executive, government administrator, scientist,legislator—is better able to decide why one courseof action might be more desirable than another.^

The fear of risk-taking

To try to eliminate risk induce a decided animusin business enterprise is against risk-taking andfutile. Risk is inherent risk-making—that is,in tbe commitment of against business enterprisepresent resources to future —in the literature ofexpectations. Indeed, the management sciences.economic progress can Much of it echoes thebe defined as the ability tone of tbe technocratsto take greater risks, of a generation ago.The attempt to eliminate For it wants to sub-risks, even the attempt ordinate husiness toto minimize them, can technique, and it seemsonly make them irrational to see economic activityand unbearable. It can as a spbere of physicalonly result in tbat determination rather tbangreatest risk of all: rigidity. as an affirmation and

exercise of responsibleThe main goal of a man- freedom and decision,agement science must beto enable business to take This is worse tban beingtbe right risk. Indeed, wrong. This is lack ofit must be to enable respect for one's subjectbusiness to take greater matter—the one thingrisks—by providing no science can affordknowledge and under- and no scientist canstanding of alternative survive. Even tbe bestrisks and alternative and most serious workexpectations: by identi- of good and serious peoplefying the resources and —and there is no lack ofefforts needed for desired them in the managementresults; by mobilizing sciences—is bound to heenergies for contrihution,- vitiated by it.and by measuring resultsagainst expectations, ^^°^thereby providing means ^^^J^ M^n^^LZ!'L!^n^^s^^'for early correction of Management: Tasks, Respami-wrong or inadequate biuties. Piacuces, copyright

aeClSlOnS. Reprinted by permission ofHarper & Row, Publishers, Inc.

All tbis may sound likemere quibbling over terms.Yet the terminology ofrisk minimization does