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Growth Rate - A Rate-of-Return Measure ofInvestment EfficiencyE. C. Capen, SPE-AlME, Atlantic Richfield CO.R. V. Clapp, Atlantic Richfield Co.W. W. Phelps, SPE-AlME, Atlantic Richfield Co.

IntroductionExcept for the oil finders' witching rods, perhaps noarea of our business has more folk remedies than themanagement of our financial affairs. The industry hasamassed numerous not-so-useful methods for assessingthe worth of capital-investment projects. Therefore, weview our task as twofold: tear down some of the old;and rebuild with something better, a concept of dollarinvestment efficiency that may be new to many readers.Why tear down anything? If our industry intends tomake headway against the cost-price currents, one sureway is to spend more wisely. Get a dollar's worth ofwork out of a dollar's worth of effort! Does this nothappen now? We do not think so. As long as peoplefeel comfortable measuring project attractiveness withpayout, profit-to-investment ratio, and other old standbys, we think improvement is possible.Our industry does not stand alone in its reliance onthe less useful guides. A 1974 report by the ConferenceBoard! shows that 14 percent of the 136 companies inthe study use payout as their exclusive yardstick. (Thecompanies that par ticipa ted in the survey covereda wide range of size and type of business.) Another59 percent use payout as one of several yardsticks., 'Simplicity and ease of understanding are among theprincipal attributes of this method according to its advocates," says the report. One wonders whether the questfor simplicity has a reasonable payout. These indices,for all their past usefulness, do not provide a managerwith the necessary guidance for using his investmentdollars most efficiently. We offer a replacement that we

call growth rate - not a panacea, but a definite step forward. The idea of growth rate has been floating aroundfor some years. We have attempted to take the thoughtsof several authors and meld them. In addition, we haveset forth some practical applications of the method andtried to outline some spots where an unwary user canget bushwhacked.Besides explaining the concept, we examine some issues that are important to its application: (1) the problem of defining "investment," (2) choosing a discountrate, (3) accounting for a variable reinvestment rate,and (4) capital allocation.In all that follows, we have left out considerations ofuncertainty. Not that risks are unimportant, but we believe that we first must understand how to deal with thefuture under the assumption of perfect certainty.Technical and Philosophical QuestionsMan still debates the relative mixture of art and sciencein the field of economics. We could devote our entireeffort here to the science side-but that would be cheating. Economics is not used in a vacuum; it must coexistin a world of diverse opinion. We think that the psychology of the economic-analysis game plays a role justas important as the manipulation of the numbers. So,now and then we devote attention to people and our perception of the influence they exert on the evaluation task.

It all starts with this diverse opinion just mentioned.Our goal may be to attach some figure of merit to aproject, but we immediately run into the fact that

What you always wanted to know about practical economics but were afraid to ask - withspecial emphasis on what to do if you have more opportunities than money. Simply choosethose projects that maximize your future worth. Start by ranking the projects with investmentefficiency, or its return counterpart growth rate.MAY, 1976 531

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people 's value scales may not be similar. von Mises 2tells us that the entire market operation proves the thesisrepeatedly. When you buy your next villa, you willhave to cough up quite a bundle of money. Examine theviews of buyer and seller. You would rather have thevilla than the bundle. The seller would rather have themoney than the villa. Because buyer and seller seevalue differently, they are able to strike a happy trade.All voluntary trades involve the same logic. The systemworks well precisely because of the opposing views ofbuyer and seller.To make an analyst 's life even more interesting, hefinds that quite a few people have very little interest inhelping him put numbers on projects-especially onthose very risky ones that prove so difficult to quantify.The conflict is not at all hard to understand. Numbersalways seem to have trouble accurately reflecting thefeelings and emotions of those closest to the project.We should not be surpr ised, then, at a reluctance on thepart of many to place faith in the "numbers."But what alternatives do we have?Recap of Common Economic IndicesPeople grow up using a cer tain economic index, and itgets to be comfortable, like an old pair of slippers.They learn from experience when it will give them goodinformation and when it may not. In our industry, itmay be 10 .years before one knows whether his indexled him astray. The necessary experience comes painfully and slowly. Even so, a capable man exertingsound judgmen t can use a relatively poor index andperhaps not come out too badly. For the benefit of thosewho do not possess the necessary experience, we wouldlike to review briefly the more common "slippers," trying to point Obit what they do well and what they dopoorly. These short descriptions are included in Appendix A, covering payout, profit-to-investment ratio, bookprofit, stockholders' rate of return, present worth, andinternal rate of return.Why so many? Campbell3 explains that "with thevariety of strategies and circumstances encountered inthe petroleum industry, no one approach offers total reliability. The practical answer is to establ ish severaltime-value criteria, use them consistently and continually check back on performance. This array should logically include payout, one or more rate of return criteria,and some method of characterizing present worth orasset appreciation."We tend to agree with Campbell when he says thatnothing gives 1OO-percent reliability. But the questionis, why? Does an index fail because the input failed,because management changed goals , or because theindex itself is built upon sand? I f one does not havereliable input data, then a whole battery of indices willnot help.We intend to show evidence why we believe the efficiency measures that follow will take the place of thatlist and, at the same time, do a better job .Investment EfficiencyPresent worth as an investment crit erion frequentlycomes under attack because it does not measure the efficiency of an investment. A project with a net presentworth of $500,000 and requiring an initial outlay of532

$200,000 looks more attractive than one with a netpresent worth of $400,000 and requiring an initial investment of only $100,000. I f we are not subject to capital rationing and these two projects are mutually exclusive (that is, we can only do one or the other), mostpeople would agree that the first one should be chosen.However, if we do have some limit on our capital expenditures in the current period, we might be veryfoolish to choose the first project over the second. Thiswould become obvious if we had a second $400,000present worth project available, so that the two projectstogether would give us a present worth of $800,000for the $200,000 initial investment.To overcome this deficiency, some experts haveproposed using present worth per dollar invested as aprofitability criterion. This measure goes by the namespresent value index, profi tabi li ty index, discountedprofit-to-investment ratio, and investment efficiency, toname just a few. We have chosen to use the last name,investment efficiency. Lorie and Savage4 were amongthe first to suggest the use of investment efficiencyunder conditions of capital rationing. Solomon5 also advocated using the "present value per dollar outlay"when comparing projects requiring different outlays.The concept has not gained much popularity in the literature of the petroleum industry, however. In fact, Silbergh and Brons6 saw no merit at all in this yardstick.What does investment efficiency have to recommendits use? First of all, as an accept/reject criterion, it isconsistent with present worth (which is generally regarded as the most suitable accept/reject criterion). Investment efficiency is just PW/I, where PW is the netpresent worth of the project and I is the discountedvalue of the required investment. Obviously, investmentefficiency will be positive when present worth is positive and will be negative when present worth isnegative.What else does investment efficiency have going forit? As a ranking criterion, it can be useful in capitalallocation. I f we are limited by a capital budget, we canuse investment efficiency to choose projects that willmaximize the present worth of the whole investmentprogram undertaken. Assume for simplicity that all ourpossible projects require investment outlays only in thecurrent year, and that we have a budget limitation onour total investments for this year. We can maximizeour present worth for that budget by ranking our projects according to inves tmen t efficiency, and thenselecting projects as we move down the ranking untilour total expenditure equals the budget constraint. Thisignores the problem of discontinuities (we may have noset of projects whose total investment exactly equals thebudget), but that is a minor problem when the individual projects are small relative to the total budget.Growth Rate of ReturnThe investment efficiency ratio does not have muchmeaning to people who are accustomed to using internalrate of return or some other rate-of-return measure. Tosay that a project has an investment efficiency of 1.2 isnot very enlightening to these folks. If we could somehow express this as a meaningful rate of r eturn , itperhaps would be more helpful. This leads us to whatwe call the growth rate of return (GRR).

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or

or

1 (B)RR = t loge I (2)

i '

projects. Growth rate of return thus gives the sameranking of projects as investment efficiency.In GRR we have a measure that (1) yields the sameaccept/reject decisions as present worth, (2) yields thesame ranking of projects as investment efficiency, and(3) is expressed as a rate of return. In addition, the calculation of GRR is straightforward. I t does not requiretrial-and-error methods, as does internal rate of return(lRR), nor does it produce multiple solutions.

Th e idea behind GRR has been around for sometime. In 1956 SolomonS used an explicit reinvestmentrate and then derived a "per annum yield promised byeach alternative course of action from its inception to acommon terminal date." This was equivalent to GRR.Baldwin7 proposed a similar measure in 1959, but without the stipulation that a common terminal date or timehorizon be used. Babcock8 formalized the concept ofGRR and analyzed it in great detail in a doctoral dissertation in 1965. Similar measures have been proposed inthe literature of the petroleum and mining industries.Phillips9 suggested an "appreciation of equity" conceptin 1965. Like Baldwin, Phillips does not use a commonterminal point in the future for all projects. Instead, hecalculates an appreciation rate to the end of a project.Otherwise, his appreciation rate and GRR are the same.He does illustrate how using a common terminal datefor two projects would change his appreciation-rateanalysis. He even hints that this might be a reasonablething to do, but does not pursue it further. In 1971,Berrylll presented a "wealth growth rate" that is identical to GRR except for the common-terminal-date assumption. He observes that calculating growth rates justto the end of each project implicit ly assumes that thereinvestment of funds at the end of the project earns atthe projec t's growth rate, rather than at the companyreinvestment rate. He argues that "management, because of its experience and learning with the projectunder question, should be able to search for and implement a r e pl ac em e nt p r oj ec t with an equivalent orsuperior wealth growth rate." III I f that is true, it seemsincorrect to be using some different reinvestment ratefor the funds generated by another project , not yet completed. We have proposed using a common terminaldate or time horizon for all projects, so that the sameassumption about the reinvestment rate on cash flows ismade for all projects. I f we did otherwise, we would

+--------------USING OPPORTUNITYRATES, COMPOUND 0ALL REVENUE(ZIFORWARO TO TIMEHORIZON. DISCOUNT INVESTMENTS BACK TOREFERENCE POINT.

+ - ' - - - - ' - ORIGINAL

1Il CASH 0FLOW

. . (3)( PW ) litGRR = i- + 1 (1 + r) - 1 ,

GRR = (BII)l!t - 1 (1)For continuous compounding,

IeGRR t =B,

for annual compounding.The continuous compounding formula derivation appears in Appendix B.We can rewrite Eq. 3 to make the relationship between GRR, r, and PW clearer:1 + GRR = ( P W + 1 ) lit1 + r I

I f PW is negative, then (PW/I + l)l/t is less thanunity and GRR < r. I f PW is positive, then (PW/I +l)l/t is greater than unity and GRR > r. Thus, GRR isequivalent to PW as an acceptl reject cri terion. Wemerely accept any project for which the growth rate isgreater than the discount rate.How does GRR relate to investment efficiency? Inusing growth rate, we will require that the same timehorizon, t, be used for all projects. Then, from Eq. 3, i tis easy to see that if one project has a higher investmentefficiency (PWII) than another, it will also have ahigher GRR. Remember, r a n d t are the same for both

How is GRR related to present worth? Appendix Bshows that if we use the same reinvestment rate, r, inall years, we have

The concept of GRR is illustrated in Fig. 1. There weshow the net after-tax cash flow resulting from a project. To calculate GRR, we first compound all the positive cash flows forward to some time horizon, t years inthe future. Any cash flows beyond that time are discounted back to that point. The rate at which we compound (or discount) these cash flows is the reinvestmentrate or opportunity rate for the company. That tells ushow much money we will have at time t, counting therevenue from this project plus the "interest" we earnby reinvesting in future projects. Let us say the total isB. We then discount the negative cash flows (investments*) to the present at the same rate to get an equiva le n t i n v es t me n t, I . We now se e that ou r projectpromises to yield us B dollars at time t if we investthe equivalent of I dollars now. If we were to put theseI dollars in the bank instead, what interest rate wouldthey have to earn for us to do as well? That interestrate is the project's GRR. It tells us that we will be aswell off investing in this project and reinvesting theproceeds at our opportunity rate as we would if we putthe money in the bank and earned at GRR. For annualcompounding,

1( 1 + GRR)t =B,

*See a more detailed discussion of investment later. Fig. 1 - Concept of growth rate.MAY, 1976 533

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-1,000750750-1,000210210210210210210210210210210

o12345678910Internal rate of return 16.5 % 31.9 %Growth rate at 10 percent 12.8 % 12.9 %

According to IRR, Project B is much better than ProjectA. Growth rate ranks Project B only slightly higher thanProject A. To judge which criterion gives a more realistic comparison between these projects, let us look at theworth of each project at the end of 10 years, assumingour reinvestment rate is 10 percent. I f the positive cashflows from Project A are reinvested at 10 percent, aworth of $3,346 in Year 10 results. I f we do the samewith Project B, the value is $3,376. These valuessuggest that there is not really much difference between

changed. Also, as an accept/reject criterion, GRR remains consistent as the time horizon changes. As longas PW is positive, GRR > r.Another complaint about GRR is that it is not as sensitive as lRR. Thus, a project that looks far superior toanother in terms of lRR may have only a slightly higherGRR. We believe that it is IRR that is misleading insuch cases. Consider two projects, Projects A and B,with the following cash flows:

Year Project A Project B

have no guarantee GRR would produce the same ranking of projects as investment efficiency.Some people object to GRR because the rate calculated for a project depends on the time horizon chosen.This is certainly true. Fig. 2 shows the GRR's for twoprojects as functions of the time horizon. The cashflows for these projects are shown in Table 1. We mightthink of GRR as a weighted average of a project 's internal rate of return and the company reinvestment rate.Fig. 2 shows that as the time horizon is moved furtherbeyond the end of a project , the reinvestment rate becomes more heavily weighted in calculating GRR. Theimportant thing to note, though, is that the relative rankings of projects do not change as the time horizon is

GROWTH RATE %Fig. 2 - Growth rate lO vs time horizon.

'''' . . . ----------------------- ,.." ' . - -------------------_____1'"

I0:" '"-

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the two projects. There is less than I percent differencein the worth generated through the 10th year. Yet, IRRsuggests that one is almost "twice as good" as theother.One might argue that, in calculating GRR, we arereally calculating the worth of a project, plus reinvestment opportunities, rather than the worth of the projectalone. That is true. But , as Solomon5 has pointed out,in ranking proposals "the valid comparison is not simply between two projects but between two alternativecourses of action." Growth rate more accurately measures the value of the alternatives.Discount RateThe calculation of growth rate or investment efficiency(as well as present worth) requires an explicit assumption as to the discount rate. Over the years, a number ofauthors have written a few thousand pages on the subject of the proper discount rate to use in discounted cashflow analysis. The inconsistencies might lead the noviceto believe that the economists do not know what theyare talking about, or at best cannot agree. People havebeen asked to use the cost of capital, a corporate cutoffrate, the average future reinvestment rate for the company, or whatever the boss says.Most authorities agree that, in the absence of capitalrestrictions, the cost of capital is the proper discountrate to use. However, investment efficiency and growthrate seem most useful in situations where capital is limited. What rate should you use then? We suggest youstick with the average future reinvestment rate. Asexplained earlier, you will want to count the sum ofmoney that you will have "down the road" from having conducted a certain project today. Therefore, youmust use a discount rate equal to your reinvestment ratefor the future. Solomon5 presented a similar view, arguing for the use of cost of capital for accept/reject decisions, but stating that "if the present value is to be usedas an index of relative profitability, the expected reinvestment rate or set of rates should be used as the discounting factor. These rates will be equal to the company's present cost of capital only by coincidence."Some will say, "What do you mean by future reinvestment rate? Are you talking about the future averageof all investments or of the future average of those investments we will make as a result of conducting thesespecific projects today?" One could argue that thosetwo future averages are one and the same, that one cannot color some dollars red and some green to trace themin and out of the company treasury. On the other hand,there are in fact marginal kinds of things that one cando later as a result of having taken a certain actiontoday. We have not come up with a satisfactory wayto test the validity of either argument. As a practicalmatter, we do not think the two rates could be verydifferent.Reinvestment rates can change from year to year,though usually not by much. Growth rate also canhandle a varying future reinvestment rate. More aboutthat later.What kinds of things happen to project select ionwhen we use cost of capital or corporate cutoff rate? Letus first make sure we mean what you think we mean bythose terms. Cost of capital would be the after-tax costsMAY, 1976

one would have to pay for all the money he needs. Usually the sources and rates vary. The bank may chargemore than the debenture holder. The cost of capital thenbecomes some weighted average for the several sourcesof money, including equity.The corporate cutoff rate is a return below which wewill not knowingly accept a project. "Knowingly" isa key word here, especially in the exploration endof the business. We feel quite sure that most dry holesfall well below anyone's minimum return criterion, butwe all have them. We really have to use the idea ofexpected value. That is, use the probability-weightedaverage of the good and evil that may befall the investor and see if that average surpasses some cutoff orminimum return. Consider the after-tax cash flows ofthree very simple projects:

Year Project C Project D Project E0 $-100 $-100 $-100I 0 45 1232 0 45 03 144 45 0

Using yearly discounting at 5 percent for cost of capital,8 percent for corporate cutoff rate, and 12 percent foraverage future return, we can construct a present worthmatrix:

ProjectDiscount C D ERate (dollars) (dollars) (dollars)

Cost5 percent of 24.39 22.55 17.14capitalCorporate8 percent cutoff 14.31 15.97 13.89rateAverage12 percent future 2.50 8.08 9.82return

We have underlined the largest value in each row toillustrate which project wins under each discount rate.Let us say we have only $100 to invest. Which of thesethree projects should we choose?I f we are cost-of-capital or iented, we will chooseProject C. Likewise, i f we have been instructed to usethe cutoff rate, we will choose Project D. If we havebeen brought up under the reinvestment-rate school, wewill choose Project E. We submit that only one of theseprojects can be best for us. It cuts no ice to argue that

the project quality varies with discount rate. Those projects have no earthly idea of what discount rate peopleare using-they could not care less.Our position has been that growth rate or presentworth efficiency will get us to the right choice-if weuse average future reinvestment rate as our discountrate. To see just why, ask the following question. Howmuch money will we have in our hands 3 years fromnow if we do one of these projects? That means wecount the money we get from the project as well as themoney we get upon reinvestment of that cash. Thesimplest way to make the calculation involves the mere535

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compounding to the third year of the positive cash flow(disregarding the initial investment). In other words, weare counting the pile of money at the end of somerainbow.

at 7 percent over its lifetime, one bought in Year 2 willearn at 6 percent , etc.Now assume we have two projects to choose between, with these end-of-year cash flows:

Now let us calculate each project's PW, using thereinvestment rate to discount the cash flows:PWF = $2,100 - $1,000 = $523.(1.05)4 (1.06) (1.07)PWG = $1,600 - $1,000 = $495.1.07

Project F looks better than Project G according to PW;but how would we stand at the end of Year 6? WithProject F, we would have a net gain of $1,100 ($2,100- $1,000 = $1,100). I f we invest in Project G andreinvest the proceeds in a 7-percent CD, we will have$1,244 [($1 ,600) (1.07)5 - $1 ,000 = $1,244] at theend of 5 years. Project G looks better than Project F.What is wrong? I t appears that the use of PW is consistent with an assumption that the cash flows generatedby a project are reinvested in projects of I-year duration, rather than in the kind of project we specified (theCD's). In our example, the PW calculation is equivalentto assuming that the $1,600 generated by Project Gearns at 6 percent for the first year and at 5 percent peryear thereafter . Under those assumptions, Project Gwould give us a net gain of $1,062 [($1,600) (1.06)(1.05)4 - $1,000 = $1,062]. Project F, having a gainof $1,100, looks better, just as with PW.We can easily use GRR to show the preference forProject G over Project F. We compound our positivecash flows forward to a common time horizon (say 6years). Of course, for Project F no compounding isnecessary. For Project G, the compounded positive cashflows at Year 6 are $2,244 [($1,600)(1.07)5 =$2,244]. Substituting in Eq. 1, we get the growth rates:

GRRF = ( 2,100) 1/6 _ 1 = 0.132 or 13.2 percent.1,000GRR, = ( 2,244) 1/ 6 - 1 = 0.144 or 14.4 percent.

G . 1,000

Project C: ($144) (1.12) = $144Project 0: ($45) (1.122+ 1.121+ 1.12) = $151.85Project E: ($123) (1.12)2 = $154.29Project E wins. It should be clear that whichever proj-ect yields the largest sum at the end of the third year

will give the largest amount for any year whatsoever,since changing the year involves multiplying each sumby the same dicscount factor. We can only calculatethat amount by using the average future return of thecompany. Cost of capital will not serve as a usefulcompounding vehicle, nor will a corporate cutoff rate.Since we are trying to get that portion caused bY'reinvestment, we must know the character of the projectsthat will come later. We cannot chase each dollar wemake to its ultimate home, so we use some averagehome or some average future project.As a practical matter; one may never know his pastaverage return (discounted cash-flow type) exact ly.How, then, will he know the future? He will not knowwith great precision. But it seems he is better off usinghis best estimate of future earning power than he wouldbe by using cos t of capiial, for ins tance, which isguaranteed not to provide much insight into our futureworth. Predicting one's future return may not be astough as it looks because competitive pressures tend tomake the future look pretty much like the past.Varying Reinvestment RateMost present worth advocates suggest that one coulduse different discount rates in different periods in calculating PW. That is about as far as they carry it usually,leaving it up to the reader to dream up his own examples and examine the implications of the procedure.We think that calculat ing PW using a discount ratethat varies from year to year can give misleading results. Porterfieldl l has expressed similar misgivings.This may be a minor problem, in that few people maybe tempted to try such a thing. But we believe that GRRcan be adapted to treat this situation correctly.Consider a simple example. Let us say we look uponour discount rate as a reinvestment rate. Further, assume that the reinvestment projects our company hasavailable are somewhat like 5-year certificates of deposit (CD's). They will earn interest (compounded annually) at some fixed rate for 5 years, at which time wecan recover the principal plus interest. Assume that weforecast the following interest rates on CD's:

YearTime 0123456

Project F$ -1,000

2,100

Project G$ -1,000

1,600

The table indicates that a CD bought in Year 1 will earn536

Year123456

Interest Rateper Annum(percent)765555

Growth rate thus indicates that Project G is betterthan Project F. Growth rate yields an answer differentfrom present worth because we compound the cash flowfrom Project G at a rate of 7 percent for 5 years, ratherthan use a new rate each year. This is consistent withour definition of projects available for reinvestment,which were like CD's.For most firms, the I-year reinvestment projects implied by PW calculations seem unrealistic. Generally,the cash generated by a project in a given year will gointo some relatively long-term projects, earning at someJOURNAL OF PETROLEUM TECHNOLOGY

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= 0.163 or 16.3 percent.

rate. These projects, in turn, will generate cash flowsthat will be reinvested in other projects to earn at ratesthat may vary, depending upon timing.Let us examine a somewhat more realistic case:ReinvestmentRateYear Project H Project J (percent)

Time 0 -1,000 -1,000 151 250 0 152 250 0 153 250 0 104 250 440 105 250 440 106 250 440 107 250 440 108 250 440 109 250 440 1010 250 440 10

Assume in this case that the cash flows are receiveduniformly over each year and continuous discountingand compounding are used.I f we use the reinvestment rate as the discount ratefor each period and calculate present worth, we get

PWH = $406.PW J = $412.

Project J looks slightly better than Project H according to PW. But now make an explicit assumption aboutthe form of the reinvestment projects. Let us say that allcash flows are reinvested in 20-year projects with uniform cash flows that, in turn, are reinvested in 20-yearprojec ts, etc. Assume that the rate of return on anyreinvestment project is the reinvestment rate shown forthe year the project begins.Under these assumptions, the benefits from Project Hwould have grown to $5,119 at the end of the 10thyear, while those from Project J would be $4,461. (Calculating these values is a bit tricky since we have tokeep track of the cash flows from all the reinvestmentprojects.) Since both projects required the same initialinvestment, Project H is considerably better.For GRR calculations, we can use Eq. 2 with a timehorizon of 10 years to getGRR H = -.L loge (5,119)10 1,000GRR = -.L log (4,461) = 0.150 or 15.0 percent.J 10 e 1,000

ment rate. The "today" dollars that one invests usuallyhave more value than the "tomorrow" dollars one getsback. Neglecting the effects of this decrease in purchasing power and its changing rate could lead to misallocation of funds.We now feel more skeptical about claims that presentworth and investment efficiency can properly accountfor variations in reinvestment rate by using differentdiscount rates in different periods. The growth-rate concept, we believe, has a definite advantage here."Investment" - The Capital ActuallyBeing AllocatedTo calculate growth rate or investment efficiency, onehas to define investment. Consider an exploratory well,for instance. Investment may include development costsif we are lucky enough to hit on the wildcat. And theremay be subsequent spending for workovers, secondaryrecovery, etc. While we will be the first to admit thatthe definition of investment for purposes of calculatinggrowth rate (or any other efficiency measure) is farfrom settled, we will share with you an example andsome heuristic reasoning. All investments do not necessarily go into the denominator for efficiency or growthrate calculations. We generally include only those thatare absolutely necessary to get the project under way.Other subsequent investments (or better, negative cashflows), economic though they are, may lead to difficulty if included.As a rule, we advise that you include in the "investment" all negative cash flow (discounted back to timezero) up until the time when positive net cash flow begins (see Fig. 3). Usually, once the cash flow becomespositive, the project begins to pay its own way in theworld and need not place a further drain on the corporate treasury. Even small negative cash flows that comelater can be "paid off" by the project so that these dipsalso put no pressure on the treasury. We use the word"investment" in two ways, conventionally and as negative cash flow. The context should make it clear whichmeaning applies.Without proving that the above rule is infallible (wedo not know that it is), let us look at an example. Weshall set up two projects, one of which is clearly betterthan the other. Yet, if we do not handle investment according to the suggestion above, the better project willnot come out with the higher growth rate.In the following table, note Years 4 and 5, the onlyones where Projects K and L differ. The question be-

+

Fig. 3 - Cash flow neluded in investment.

Do notInc lude in"Inves tment"

Time _

}-III~ I

: Inc lude in :: " Inves t - :: ment" :I II I:..I I

.....

. c : k - - - - - ~ < - - - - ~ ~ < - - - - - - - - - - =tilttlC)

Growth rate correctly ranks Project H ahead of Project J.Such situations probably do not arise often. It is difficult enough to pick a reasonable reinvestment rate touse in the future, without having to predict year-to-yearvariations. But we can think of possible applications. Amajor shift in emphasis by a company through good fortune, bad luck, or a merger could lead to a sudden shiftin corporate reinvestment rate. Also, variations in therate of growth of the money supply, along with otherfactors, lead to price movements that can affect projecteconomics as well as cause changes in one's reinvest-MAY, 1976 537

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comes very simple. Would you rather have $100 thisyear and nothing next year, or do you prefer to get $200this year with the stipulation that you must return $100next year? You would take the second choice, wouldyou not? The net difference in the choices is that thesecond allows that we give you $100 now and you return $100 a year from now. About the worst that couldhappen (in an expected sense) would be that you putthe money in a savings bank and draw interest for yourself without having to pay us any. We say then thatProject K certainly looks better than Project L becausethose 2 years give Project K the higher present worthfor exactly the same initial investment.

Project K WithYear 5 NegativeCash FlowDiscounted toTime a andIncluded inYear Project K Denominator Project La -1,000 -1,062 -1,0001 100 100 1002 200 200 2003 300 300 300- - - - - - - - -4 200 200 100 I5 -100 a a I- - - - - - - -6 200 200 2007 300 300 3008 400 400 4009 200 200 200

10 100 100 100PW IO ,dollars 132.92 132.92 126.71GrowthRate 10 ,percent 11.38 11.30 11.32BabcockB and Berrylo recommend making Project Klook like the second cash-f low column with the fifthyear discounted investment put in at time zero. Thatmakes the growth rate come out a little less than it doeson Project L. You know that cannot be right. Youwould never take Project L over Project K. We believe

that the growth rate for Project K should come outhigher or the whole concept is in trouble. By adheringto the suggestion outlined previously, we do get the bestproject to have the higher growth rate, as shown in thefirst column of the table.One might fairly ask why we have concerned ourselves with what seems to be such a piddling difference-growth rates that differ only in the seconddecimal. First, we do not know that the differences willalways be so small. Second, in a highly competitiveenvironment, every little bit helps.Why would we logically want to include the benefitsof future investments if we were not willing at the sametime to use those investments in the rate-of-return base?To understand, let us try a different way of looking atProject K above.

Project K IncrementalWithout Effects ofYear 5 Year 5Year Project K Investment Investmenta -1,000 -1,000 a1 100 100 a2 200 200 a3 300 300 a4 200 200 a5 -100 200 -3006 200 200 a7 300 100 2008 400 100 3009 200 50 150

10 100 50 50PW IO atTime 0,dollars $132.92 -$6.28 $139.20Growth rateat timeof firstinvestment,percent 11.4 9.9 23.0Project K without its fifth-year investment and addedincome would not be very attractive. Yet if we had notspent the $1,000 to get us into Project K in the firstplace, we would not have had the opportunity to make

TABLE 2 - PROJECT RANKINGS FOR CAPITAL ALLOCATIONProject

P Q R S T U V W X y ZMeasure

Profit/investment 7 8 5 4 10 9 3 2 11 6 1Payout 5 4 7 8 3 2 9 10 1 6 11Book profit 5 4 9 10 2 1 11 8 3 6 7Internal R/R 3 2 11 10 5 1 9 8 4 6 7Present worth at 5 percent 8 9 4 2 10 7 1 6 11 3 5at 10 percent 4 2 9 6 10 1 5 8 11 3 7at 15 percent 3 2 11 10 4 1 9 8 5 6 7Growth rate at 5 percen t 8 9 6 4 10 7 3 2 11 5 1at 10 percent 6 4 9 8 10 3 7 2 11 5 1at 15 percent 3 2 11 10 4 1 9 8 6 5 7Investment efficiencyat 5 percent 8 9 6 4 10 7 3 2 11 5 1at 10 percent 6 4 9 8 10 3 7 2 11 5 1at 15 percent 3 2 11 10 4 1 9 8 6 5 7Democracy 5 2 11 8 9 1 7 6 10 4 3538 JOURNAL OF PETROLEUM TECHNOLOGY

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all the investments , so clearly we must find some wayto decide which projects merit our attention for the cashat hand.Pro ject Z shows the highest profi t- to -inves tmentratio, Project X shows the shortest payout, Project Ushows the highest internal rate of return and largestearly book profit, and so on. I t might be worthwhile tolook at Table 2, where we show the ranking of theprojects by each of the economic indices. Project P,for example, comes in seventh on profit-to-investmentratio, fifth on payout, third on internal rate of return,and sixth on growth rate at 10 percent.The more democratically inclined find these tablesextremely useful. Rather than choose a single index tolive with, they would rather look at all of them andsomehow average the results. To help those who choosethat route, we have averaged the rankings for eachproject and created a new ranking in Table 2, labeledsimply "Democracy." Note that Project U wins thehonors.How shall we decide which projects are the best forour money? One way involves counting the futuremonies. I f we choose Project P today, jus t how muchmoney will we have 10 years from now that we can attribute to that project? (We could look at any timehorizon; we chose 10 years because it was handy.)Many will agree that we should do those projects thatgenerate the most value over time.Firs t, how do we calculate how much money we willget from our efforts? Choose Project Y, since its cashflow picture is so simple. We start by spending $1,000.The next year we have no investment , and find that ournet proceeds after tax and after all costs comes out tobe $210. Rather than stash that money away, we willreinvest it.And when we reinvest, we expect to make some return. So all the dollars we make from Project Yarebeing reinvested as we receive them into other companyprojects at the average company reinvestment rate,which we assume to be 10 percent. Year 1's proceedsbecome $495 [$210 (1 + 0.10)9 = $495] at the end of10 years. Year 5' s proceeds become $307 [$210 (1 +0.10)4 = $307], and so on. We simply compound eachyear's cash forward to the time horizon and sum.At the 10th year, Project Y provides a pile of moneyof $3,346. Similarly, Project P gives $3,331 in 10years. We go down the list choosing those projects thatgive the largest amount in 10 years per dollar invested.

that fifth-year investment. So why not give the initial$1,000 all the benefits it makes possible? When it istime for the fifth-year investment, we are quite surethose in charge will examine it again to make sure itwill perform as advertised. I f it will not, then perhapsthat money will go elsewhere. In that event , our original $1,000 does not come out so well. But we shouldremember that faulty input to blame and not thecalculation procedure; that is, we did not attach theproper risk to the return from the fifth-year expenditure.We think that the whole idea of investment as itrelates to an efficiency measure deserves more work.Present ly, the definit ion seems arbitrary. For allocation purposes, perhaps oj;e need worry only aboutthat money he will have to come up with during theproject startup period. That is, measure the efficiencyof those dollars for which other projects are currentlycompeting.Capital AllocationWe have gone through the preliminaries and must nowproceed to the finals-the payoff. None of what wehave discussed will help anyone unless he puts it to use.We call the testing ground "capital allocat ion," orwhere the action is. Few people have more money thanprojects. The more general problem consists of a manager, a staff recommending projects designed to break abank, and a stingy treasurer who comes from five generations of misers and wants to see justifications on theuse of paper clips.How should we allocate funds in such situations?Ideally, we believe company management would preferto instruct some genie to line up all the projects on oneside of a big room and all the money on the other side,and then choose that set of projects that does not exceedthe budget but maximizes the worth of the companyover time. As most every manager knows, genies likethat are almost impossible to hire. We have to resortto humans and whatever weaknesses humans bring tothe problem.Several divisions may be vying for the cash. Eachwill try to make its own projects look superior. That is,the playing of politics sometimes has more to do withhow funds are spent than do the economics. We mustassume that managers have honorable intentions andthat each makes reasonably good estimates of investment, timing, and profits for his projects.To illustrate the problems that confront people in capital allocation, let us dream up some projects withgreatly differing cash-flow patterns. We calculate several of the well known economic indices and see whatthey lead to. (Refer to Table 1.) To simplify, assumethat each of the II projects requires only one investment, and that is required at time zero. Further, eachproject will be considered as having a lO-year length(though a glance will show that Project X gives noafter-tax money beyond Year 8).Below each project we find profi t- to-investmentratio, payout, early book profit, internal rate of return,present worth (three discount rates), growth rate (threediscount rates), and investment efficiency (three discount rates).We have only $3,000 and must choose among these.11 projects. We would need a full $10,000 to get intoMAY, 1976

ProjectP#QRST

#UV#WXY#Z

Future Worths of ProjectsFuture Future Worth perWorth Dollar Invested(dollars) (dollars)3,331 3.3313,356 3.3562,989 2.9893,190 3.1902,982 2.9823,431 3.4313,293 3.2931,784 3.5682,955 2.9553,346 3.3461,836 3.672

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The # 's indicate the projects that would be chosen ifmaximizing the pile of money in 10 years were ourcri terion for project select ion. (We hope it is clear thatbecause of the compounding and discounting, the projects that give the largest pile of money in the 10th yearalso give the largest for any other year.) Our original$3,000 buys us four projects, starting with the two $500ones. The projects in order of their attractiveness are Z,W, U, and Q.That ranking turns out to be identical with what wewould have determined had we used growth rate orpresent worth efficiency assuming an average opportunity rate of 10 percent. Since growth rate and efficiencyare based on maximizing future worth , we should notbe surprised.It might be interesting to see what rankings other investment criteria would show for the $3,000 and howbig a bank roll they would have helped us generate .

GRR identif ies projects that give the most money atthe 10th year. The more common indices cannot keeppace.Many people incorrect ly surmise that one can properly rank projects with internal or discounted cash-flowrate of return. According to Mathur and Carey, 12 amajority of major petroleum companies use IRR as theirfirst preferences for profitability measure. Presumably,they make some choices and comparisons. The Conference Board report! shows that about 65 percent of thecompanies use that one measure, with 15 percent usingit exclusively. One manager, explaining his company'sheavy use of the index, said, "Also, it is an excellentdevice for the comparison of alternative investments."The index, when used on the dif fe rence in cashflows, will compare two projects properly. However, torank a long list of projects by such pairwise comparisons would be quite a chore.We have discussed the discoun t rate and how tochoose it. I f one uses growth rate or efficiency butchooses the wrong discount rate, he could be in fortrouble. I f he wanted to go the cost-of-capital route oruse a corporate cutoff rate, he would find himself withProjects Z, W, V, and S for $10,103, somewhat lessthan the maximum possible.Another possible miscue comes from the fellow whothinks he is making a higher return than he really is.Consider the man who knows that his superior managerial abil ities are bringing in a IS-percent return for hiscompany when, known only to the little birdie, he isonly making 10 percent. He will choose Projects U, Q,

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CriterionProfit/investmentPayoutEarly book profitInternal rate of returnPresent worth at

10 percentGrowth rate at10 percentDemocracy'One-half project to use up $500.

Projectsin Orderof ChoiceZ, W, V, SX,U,TU,T,XU,Q,PU,Q,Y

Z, W, U, QU, Q, Z, Y/2*

Pile of Moneyat the 10thYear (dollars)10,1039,3689,36810,11910,13410,40810,297

and P (highest growth rates at 15 percent) and end upwith $10,119 at the end. Some people express shockover the fact that an unrealis tical ly high discount ratemay cost us dollars in the end. Were it not so, we couldall use a 100-percent discount rate and get rich.Notice that we have said nothing of using a cutoffrate such as "accept all projects with a growth ratelarger than 10 percent." Our opportunities come in allshapes and sizes. I f our future average opportunity rateis 10 percent, it will be because we will have someprojects that make 15 percent, some maybe 20 percent,and some less than 5 percent. We, of course, do notalways know ahead of time what we will make-which

is why we get a dry hole now and then. But if ourbeforehand guesses are at least correct on the average,we will have to be accepting some projects that haveless than a lO-percent growth rate, or we fill find itimpossible to average out 10 percent in the future. Ourscheme calls for ranking the projects by growth rateor eff ic iency and accepting them one by one un ti lthe money runs out-which may mean we accept an8-percent project. Of course, there would be somelower acceptance limit. We would not want to acceptprojects at a lower return than our cost of funds.As a practical matter, one can average 10 percenteven though all the projects he allocates money for calculate at bet te r than 10 percent. He may not includecorporate or division overhead in his project calculations. Often, he has a bunch of sunk costs that he nolonger considers. So he calculates a project at 12 percent when, if he had gone back to do a "cradle-tograve" analysis, he would have come up with maybe 7percent. As mos t economists advise, however , we donot go back and cry over the spil led milk (sunk costs) .We do our best from this time forward.Also, there may exist a selection bias that causes theactual return to be somewhat lower than the calculatedreturn, even when our calculations are correct on theaverage. 13No matter how hard one might argue for maximization of worth, other goals are going to creep in. Aslong as Wall Street acts as it does, there will be somewho will strive for book profits. One could then put aconstraint into his allocation scheme. Maximize worthof the corporation under the constraint that we mustshow a certain book-profit growth. We no longer canuse growth rate by i tsel f because the measure does notconsider the "book" entries. But the efficiency idea stillworks. Line up all your projects and all your money.Look at all combinations of projects that do not exceedthe budget and that achieve the book-profit goals. Fromthat set, choose the combination that has the highestpresent worth.I f you have 200 projects to choose from, you willhave 220 0 total combinations to look at. Tha t is about1.6 x 1060 ; if you have a very fast computer and canexamine the economics of one combination per second(that means complete economics of 100 projects), thenthe chore will only take about 5 x 1052 years. For somehelp along those lines, refer to Robertson et al. 14 whosecapital allocation model will yield maximum spendingefficiency under whatever constraints you choose. Andit will do the job in just a few minutes .Or, there is a quick and dirty way to get most of the

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way there. Every manager who lines up his suite ofprojects for the year is going to have some that he justknows are worth doing and some he knows are notworth doing. And let us assume he is correct on bothcounts. His problem stems from that set of projects thathe is not sure about, but for which his budget allowssome spending. He could calculate the benefits of allthose projects he is sure he is going to do. Then hecould order the gray projects by growth rate. Startingwith the highest-efficiency project, he puts them intothe budget one by one until he runs out of money orexceeds some constraint. I f it is the first, he is finished.I f it is the second, he starts to juggle a little. Pretty soonhe will have a satisfactory combination (that is, almostoptimum).Complications-Incremental InvestmentsA more or less blind use of investment efficiency orgrowth rate can lead to a predicament. Say we have avery simple allocation problem involving a total budgetof $1,200 and three projects. To disturb matters a bit ,one of the projects, Project M, has an alternative investment plan, Project M-alt. Below we see the PW's,investments, and GRR's for all projects.

Project Project Project ProjectM M-alt N 0--- --- --- ---PWIO , dollars 100 110 15 5Investment,dollars 1,000 1,200 200 200GRR 10 , percent 11.1 11.0 10.8 10.3The largest GRR goes with Project M. One mightargue, however, t ~ a t he would rather recommend doingProject M-alt for ah 11 percent return than do Project Mand then have to drop down to Project N with its 10.8percent return to spend the rest of his budget.To help us think our way through the dilemma, weshould treat the incremental cash flow between ProjectsM and M-alt as a separate project and see how it stacksup against competing demands for money. Project M-altrequires an extra $200 investment and brings an extra$10 in PW. The GRR of the increment is 10.5 percent.Now, we can more clearly see the relation betweendoing Project M-alt and Project N. Since Project N gavea GRR of 10.8 percent, we know we should do ProjectsM and N rather than Project M-alt for our $1,200. Acheck of the combination PW for Projects M and N together, $115, proves that Project M-alt (with PW ofonly $110) is not the best place to put the investment.The example teaches us to design the project list sothat investment dollars appear no more than once. Thesecret is to layout our investment opportunities usingincremental economics to describe alternative strategiesfor doing individual projects. To put Project M andProject M-alt in the same list causes confusion because both include the same $1,000 investment and the$100 PW that goes with it.As a practical matter, one would have difficulty looking at every single incremental investment . For instance, in analyzing exploratory-well economics, somethought is given concerning the spacing for the subsequent development wells. An 80-acre sp:;lcing maycause a more efficient expenditure, while a 75-acreMAY, 1976

spacing may cause a higher PW (just as Projects M andM-alt above). But looking at each additional development well as a separate incremental investment wouldprove too much of a burden for most people. Payingattention to such details will help to increase the worthof the company through more efficient investment, butwe are nevertheless mindful that one Can spend moremoney on a fancy analysis than he can make from theproject he is analyzing. Our only advice would be theobvious-exercise sound judgment.ConclusionsMany investment criteria commonly used in the pasthave not provided very effective guidance for spendinginvestment dollars most efficiently. Discounted cashflow techniques, such as present worth and internal rateof return, represent improvements over some of thesimpler measures and are becoming widely accepted,though perhaps not always understood.In capital-budgeting situations, where one suffersfunds limitations, even present worth and internal rateof return fall short of what the decision maker needs.Neither will give proper comparisons among projects.Investment efficiency and growth rate provide rankingmeasures that can help in maximizing the worth of thefirm by making sure the best projects get to the top ofthe list.Growth rate has an additional advantage, in that itcomes packaged as a rate that is precisely analogous tointerest rates paid by banks-a concept familiar to all.AcknowledgmentThe autho rs thank their many friends in AtlanticRichfield Co. who constructively criticized the conceptof growth rate.References1. "Capital Investment: Appraisals and Limits," Report No. 641,The Conference Board, Inc., New York (1974).2. von Mises, L.: Human Action-A Treatise on Economics, HenryRegnery Co., Chicago (1966) 204.3. Campbe ll, J. M.: Petroleum Reserl'Oir Property Emluation, Petroleum Publishing Co., Tulsa, Okla. (1973) 118-119.4. Lorie, J. H. and Savage, L. J.: "Three Problems in Rationing

Capital," The Management (!t' COI]Jorate CalJital, Free Press,dist. by Macmillan Co., New York (1959).5. Solomon, E.: "The Arithmetic of Capital Budgeting Decisions,"The Management ot' Corporate Capital, Free Press, dist. byMacmillan Co., New York (1959).6. Silbergh, M. and Brons, F.: "Profitabil ity Analysis-Where AreWe Now?," 1. Pet. Tech. (Jan. 1972) 90-100.7. Baldwin, R. H.: "How to Assess Investment Proposals," Har\'ard Bus. ReI'. (May-June 1959).8. Babcock, G. c.: "Growth to Future Value as a Measure of Investment Worth," PhD thesis, U. of California, Los Angeles(1965).9. Phillips, C. E.: "The Appreciation of Equi ty Concept and ItsRelationship to Multiple Rates of Return," J. Pet. Tech. (Feb.1965) 159-163.10. Berry, C. W.: "A Wealth Growth Rate Measurement for CapitalProjects," Decision-Making in the Mineral Industry, CanadianInstitute of Mining and Metallurgy, Montreal (1971). .11. Porterfield, J. T. S .: IIlI'estment Decisions and Capital Costs,Prentice-Hall Book Co., Inc., Englewood Cliffs, N. J. (1965).12. Mathur, S. B. and Carey, O. L.: "Economic Decision MakingPractices in the U.S. Petroleum Industry," paper SPE 5011 presented at the SPE-AIME 49th Annual Fall Meeting, Houston,Oct. 6-9, 1974.13. Brown, K. C.: "A Note on the Apparen t Bias of Net RevenueEstimates for Capital Investment Projects," J. ot'Finance (Sept.

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1974) 1215-1216.14. Robertson, J. L. , Wood, D. R., and Blackburn, C. R.: "RealTime Capital Allocation," paper presented at the II th AmericanMeeting of the Institute of Management Sciences, Los Angeles,Oct. 1970.15. Metz, R.: "Per Share Earnings Concern Accountants," @1973New York Times News Service, appeared in Dallas Morning

Nell's, Dallas (July 14, 1973) Sec. D, 7.

APPENDIX APayoutPayout for an investment is simply the time required forthe projec t to generate sufficient revenue above theoperating expenses to equal the investment. Payout timeserves as an important economic yardstick. It probablyattracts the attention of smaller producers more thanmajor oil companies. Regardless of the size of the company, no manager wants to have all the assets investedin long-term projects, nor does he want them all inshort-term projects. He wants a mixture.

The smalle r firm generally uses bank financing ,where it pledges the particular asset. Most banks hesitate to make loans extending longer than 3 years. Theimportance of payout depends on what financing technique you use and what country you are operating in.An advantage of using payout time as a guide in arisk situation such as operating in a foreign countryshould be obvious. One does not have to know theequipment life, but rather the probabilities of being ableto operate for different periods of time.Probably the greatest weakness of payout is that noconsideration is given for cash flow that the projectgenerates after the payout period ends. It also ignoresthe timing of cash flows during the payout period.Profit-to-Investment RatioThe profit-to-investment ratio (P/I) is a ratio of the project 's total profit to the total investment and probablycame into being because of its simplicity. This PII ratiocan be calculated either before or after taxes.The PII ratio has one problem; the timing does notcome into play, thus destroying the effectiveness of theratio. This investment tool has just about seen its day.Book ProfitOne determines a project's book profit by taking theoperating revenue and subtracting expenses, taxes, anda portion of the investment commonly known as writeoffs. These write-offs represent some part of the original investment and part of the subsequent investment.Companies write off investments over time according tovarious accounting practices.Write-offs beget problems. Managers are often pressured to capitalize as much of the expenditure as possible, which really defers the recognition of an expense toa later t ime. It should be obvious, for a given revenueand operating cost, that book profit moves in a dollarfor-dollar relationship with book write-offs. Removing$1 of write-off increases book profit by $1 for a givenperiod.Companies are striving for higher and higher bookprofits. The "quarterly report" underlines this desire.Every 3 months, listed companies report to stockholdersthe book profit for the last quarter, comparing the re-542

suIts with those of a year ago. Naturally, to keep anupward trend, companies want each quart er' s profithigher than the year before. Consequently, some companies may place undue emphasis on the near-termbook profit.Because of the arbitrariness in write-off rules, there isa movement by several CPA groups to adopt a methodof reporting earnings whereby one gives a range ofearnings rather than an absolute value. is Such a changeprobably will be a long time in coming.Stockholders' Rate of ReturnThe average rate of return over the life of the project iscalculated by dividing the total book profit for the project by the sum of all years' average invested capital.Decreasing the original investment by the annual netcash flow will not give the average invested capital , assome people bel ieve. To determine average investedcapital in any year, one adds the original investment toany subsequent investments and then subtracts thewrite-offs (same write-offs used to calculate the bookprofits for an equivalent period). To get an average, youwould simply add the beginning-period balance to theending-period balance and divide by two.People use this rate of return because of its easeof calculation. The strengths and weaknesses of stockholders' rate of return parallel those of book profits.Present WorthOne of the most acceptable investment criteria is thepresent worth or present value concept. Present worthmay be thought of as being the value, at a referencedate, of some future cash flow discounted at a givenrate (or past cash flow compounded at the same rate).Many authorit ies agree that present worth is the bestcri terion to use in cases where there is no capital rationing. They gene rally recommend d iscounting at thef irm's average cost of capital . Projects with posit ivepresent worth should be accepted and those with negative present worth should be rejected.In most firms, however, there are capital restrictionsimposed, so that not all projects with positive presentworth will be accepted. Projects must be ranked inorder of desirability, and present worth alone is notsatisfactory for ranking under these conditions.Internal Rate of ReturnInternal rate of return or investors' rate of return (IRR)is the interest rate or discount rate that, over time, willreduce the net cash flow to zero. People like this investment criterion because they need not choose a discount rate. Several methods are used to obtain IRR,none of which are exact. Also, for some typical cashflow patterns, there can be several solutions for theIRR-a somewhat unsettling development.Fig. 4, a PW profile for two projects, shows thegraphical solution for IRR. Project V has an IRR of13.8 percent and Project U has an IRR of 28.5 percent-where the curves intersect the "0" present-worthline. (Table 1 shows the cash flows for these two projects, along with others.) Though 28.5 percent is certainly greater than 13.8 percent, one can easily show thatProject U may not be the better project.Such problems in ranking projects by IRR have been

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-20-0.9DISCOUNTRATE

252015

~ J : t ; ~Reinvestment Rate=10% ! f i ~ ~ ~Time Horizon=10 Years

/V/ ; //

./ 'V// ' ~.---/.......

3456789Investment Efficiency

Fig. 4 - Graphica l solut ion for internal rate of return.

pointed out by many authors. Others have suggestedchoosing between two projects by calculating IRR onthe difference between their cash flows. The project requiring the larger investment is chosen if its incrementalIRR exceeds the cost of capital. Besides being cumbersome, this yields nothing more than the same ranking aspresent worth.APPENDIX BWe want to show how growth rate of return (GRR) isrelated to present worth (PW). For annual compoundingand discounting, the equation for GRR is

1(1 + GRRy =B, (B-1)where

I = discounted investmentt = number of years to time horizonB = compounded value (at the time horizon) of thepositive cash flows.

We can also writePW = B(1 + rrt - I, (B-2)

wherer = discount rate (expressed as a decimal fraction).

Rewriting Eq. B-2, we haveB = (PW + I) (l + r y.

Substituting in Eq. B-1 givesI( l + GRRy = (PW + I) (l + r) t(l + GRRl = ( P ~ + 1) (l + r l

( PW ) li t1 + GRR= -1-+ 1 (l + r)

MAY, 1976

Fig. 5 - Graphical method for obtaining growth rate frominvestment efficiency.

GRR= ( P ~ + 1) li t (1 + r) -1 .For the case of continuous compounding and discounting,

IeGRRt = B , (B-3)PW = Be-rt - I, (B-4)

orB = (PW + I)ert .

Substituting in Eq. B-3 givesIeGRRt = (PW + I)eTteGRRt = ( + 1) ert

GRRt = loge ( P ~ + 1) + rtGRR =+oge ( P ~ + 1) + r.

A simple graphical method can be developed for obtaining growth rate from investment efficiency, as in Fig.5. The plot is on semilogarithmic paper, but the twocycle logarithmic scale (0.1 to 10) has -been modified bysubtracting 1.0 from each value on the scale. Thegrowth rate as a function of investment efficiency forcontinuous compounding and discounting is a straightline on this graph. The curve for an annual compounding and discounting case is also shown. In both cases,the reinvestment rate was assumed to be 10 percent andthe time horizon 10 years. JPT

Original manuscript received in Society of Petroleum Engineers office Aug.10,1973. Paper accepted for publication Feb. 11, 1974. Revised manuscript received Dec. 23. 1975. Paper (SPE 461.3) was first presented at the SPE-AIME 48thAnnual Fall Meeting, held in Las Vegas, Nev., Sept. 3D-Oct. 3, 1973.@ Copyright1976 American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc.

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