St Petersburg Paradox and Tech Stocks 2000

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The St. Petersburg Paradox and the

Crash of High-Tech Stocks in 2000

Gbor J. SZKELYand Donald St. P. RICHARDS

During the late 1990s high technology growth stock prices were

raised to unprecedented levels by avid stock purchasers aroundthe world. In early 2000, share prices subsequently underwentprolonged declines, leaving many purchasers with devastatinglosses. This article reviews some aspects of the history of theSt. Petersburg paradox and some related games. We recount aremarkable article by Durand in which the valuation of growthstocks is related to the St. Petersburg paradox. Our conclusionis that the run-up in stock prices in the late 1990s and the subse-quent declines in 2000 could have been avoided by an analysisand application of the St. Petersburg paradox.

KEY WORDS: Alan Greenspan; Cleveland Indians base-ball team; Fair game of chance; Geometric distribution; High

technology stocks; Irrational exuberance; Mark Twain; Mutualfunds; Utility function; Valuation of growth stocks.

1. INTRODUCTION

In the early 1700s Nikolaus Bernoulli formulated a problemin the theory of games of chance, now known as the St. Pe-tersburg paradox. At that time, Bernoulli was a relatively newdoctor of jurisprudence, having written in 1709 a dissertation

at the University of Basel, Switzerland, on the application ofthe calculus of probability to questions arising in the practiceof law. In 1731 he was appointed professor in law and he laterserved four times as Rector of the University of Basel. (As iswell known, the Bernoullis were a renowned family of Swissmathematicians, several members of which studied probabilitytheory; particularly notable for his work in that field was JamesBernoulli, the father of the first law of large numbers.)

On September 9, 1713, Bernoulli communicated his problemby letter to Pierre Reymond de Montmort, and this was fol-lowed by a series of letters between them. Montmort himselfwas perplexed by the problem, and he later published his cor-respondence with Bernoulli in the second edition of his bookon games of chance (Montmort 1713). There, the world couldread about Bernoullis paradox for the first time, and since thenthe problem has fascinated some of the worlds great intellects,

Gbor J. Szkely is Professor, Department of Mathematics and Statistics,BowlingGreen StateUniversity, BowlingGreen, OH 43403-0221,andthe RnyiInstituteof the Hungarian Academy of Sciences, Budapest,Hungary. Donald St.P. Richards is Professor, Department of Statistics, Penn State University, Uni-versity Park, PA 16802-2111 (E-mail: [email protected]). We are gratefulto theeditor,a referee, andSndor Csrgo forcomments, on an earlier version ofthis article, which led to a much improved revision. 2000 Mathematics SubjectClassification: Primary 60F05; Secondary 60G50.

for it was clear that standard mathematical approaches to thisproblem were not in harmony with commonsense reasoning.

In a letter to Bernoulli written on May 21, 1728, and mailed

from London to Basel, Gabriel Cramer, originator of Cramersrule for solving systems of linear equations, rephrased the prob-lem for simplicity from one involving dice to coins. Then in1738 Nikolaus cousin, Daniel Bernoulli, presented to the Im-perial Academy of Sciences in St. Petersburg an article that an-nounced the paradox to the world. In his historically memorablearticle Daniel Bernoulli also proposed a solution to the paradoxand, although the paradox was first announced to the world byMontmort (1713), the problem has come to be known as the St.Petersburg paradox.

This article reviews some of the history of attempts to re-solve the St. Petersburg paradox and we recount some related

problems, including modifications to the original paradox. WediscussindetailthefascinatingarticleofDurand(1957)inwhichtheSt.Petersburgparadoxisappliedtotheproblemofestimatingfair valuations for the shares of growth companies. Our conclu-sion is that recent extreme declines in stock prices was foretoldby Durands application of the St. Petersburg paradox.

2. THE PARADOX

LetXdenote a players net profit in a game of chance takingplace in a casino. Then the game is called fair if E(X), theexpected value ofX, is zero. Intuitively, a game is fair if, after alarge number of independent repetitions, neither the player nor

the casino has an advantage over each other.As Feller (1945) observed in an extensive discussion, this def-

inition of a fair game usually applies only if the variance ofXis finite. Otherwise, there will exist examples of fair games inwhich, with probability tending to one as the number of inde-pendent repetitions tends to infinity, the player will sustain asteadily increasing loss having order of magnitude smaller thanthe number of repetitions. Nevertheless, at least for the moment,we shall proceed according to this definition even if the varianceofXis not finite.

The St. Petersburg Paradox:Peter tosses a fair coin repeatedly until it showsheads. He agrees to pay Paul two ducats if it shows heads on the first toss, four

ducats if the first head appears on the second toss, eight ducats if the first headappears on the third toss, sixteen if on the fourth toss, etc. How much shouldPeter charge Paul as an entrance fee to this game so that the game will be fair?

In this game, the potential payoff doubles after each toss; werefer to games of this type as St. Petersburg games. In DanielBernoullis later statementof theparadox, thepayoff for k tossesis2k1 ducats, but we have chosen to work throughout with apayoff of2k ducats in order to simplify calculation of expecta-tions.

To determine the amount Peter should charge Paul as an en-trance fee so that the St. Petersburg game will be fair, we cal-

2004 American Statistical Association DOI: 10.1198/000313004X1440 The American Statistician, August 2004, Vol. 58, No. 3 225


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culate Pauls expected payoff. Surprisingly, the game cannot bemade fair, no matter how large the entrance fee.

To show this, we calculate Pauls expected payoff as follows.Letkbe any positive integer; then the probability that the gameends at thekth toss is2k, at which time Peter will pay Paul2k

ducats. Let YdenotePeterspayout; then Y is a random variablewith possible values 2, 4, 8, 16, . . .,and P(Y = 2k) = 2k, k=1, 2, 3, 4, . . .. Equivalently,log2 Yhas a geometric distributionwith probability of success 1/2. Therefore the expected value

ofY is

E(Y) =1

2 2 +

1

4 4 +

1

8 8 + = 1 + 1 + 1 + , (1)

proving that Yhas infinite expectation. This shows that no finiteamount of money can be a fair entrance fee.

In short, Paul should be willing to pay an infiniteprice to enterthis game. However, this is a requirement to which almost norational person would agree or be able to satisfy. Although thecalculation of Pauls expectation is mathematically correct, theparadoxical conclusion wasregarded by many early probabilistsas unacceptable.

Indeed, Nikolaus Bernoullis motivation for formulating thisproblem stemmed from his study of games of chance. In itsearly days, the theory of games of chance concentrated heavilyon notions of fairness, equity, and examples of fair games, andso Bernoulli was surprised and concerned by his example ofa game in which the price of entrance appeared to be grosslyunfair. As evidence of this unfairness, Paul is certainto win afinite amount, so it seems unfair to impose an infinite entrancefee. Moreover, it is highly probable that Paul will receive onlya small payment, so that even a large, finite entrance fee seemsunfair. For instance, the probability that Paul receives at most 32ducats is31/32, or 96.875%, and so an entrance fee of 10,000

ducats, say, seems unreasonable.

3. SOME ATTEMPTS TO RESOLVE THE PARADOX

Several probabilists suggested modifications to the St. Peters-burg paradox in efforts to obtain acceptable solutions. For themost part, these modifications employed the concept of utilitywhich may be defined for our purposes as follows (see DeGroot1970, p. 90).

Consider a game in which a player can receive a reward fromR, a set of possible rewards. We assume a basis of preferencesamong rewards, so that a player can specify a complete orderingofR. For simplicity, we suppose thatR is a countable subsetof the real line such that the players ordering of rewards is thenatural ordering on the real line. Regarding R as a probabil-ity space, the players preference among rewards in R leads topreferences among the probability distributions on R. IfPandQare probability densities on R, then we writeP Qif theplayer prefersQto PandP Q if the player does not preferP toQ.

For any probability density function P on Rand any real-valued functiongon R, we use the notation

EP(g) :=

rRP(r)g(r).

A real-valued function U on R is called a utility function if,for any probability distributions P andQ on R such that bothEP(U) and EQ(U) exist, P Q if and only if EP(U) EQ(U).

Nikolaus Bernoulli, in his letters to Montmort, proposed sev-eral solutions to the paradox. Bernoulli proposed the assignmentof zero probability to large values ofk in (1), thereby truncat-ing the series. In this context, Bernoulli is arguing in favor ofa utility function Ufor which U(k) = 0for sufficiently large

k. Bernoullis argument was that very large values ofk weremorally certain not to occur so that, from a practical stand-point,one loses nothing by truncating (1).Still, Bernoulli did notfind his own argument convincing, hence he sought the views ofothers.

Gabriel Cramer, the Comte de Buffon (well known for Buf-fons needle), and others viewed the calculation of expectationin (1) as unacceptable in that it simply added the actual pay-offs weighted by the corresponding probabilities. Cramers ba-sic idea was that the larger the size of a persons fortune, thesmaller the moral value of a given increment in that fortune.Cramer proposed alternatives under which the value of a sumof money is measured through various utility functions, such asthe square root, or the inverse of the amount, or by placing alimit on payouts, all of these devices being chosen so as to leadto a finite expectation. In adopting these methods, these authorsalso were proposing as a basic principle that additional sums ofmoney have decreasing marginal utility.

For example let us suppose, like Buffon and Cramer, that theutility of money ceases to increase beyond a certain amount,say, one million ducats. Or equivalently, let us make the naturalassumption that Peter has limited resources, in which case henecessarily must place a limit on the size of his payouts. Bearingin mind that219


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Table 1. Buffons Simulation of the St. Petersburg Game

Tosses (k ) Frequency Payoff ( 2k)

1 1061 2

2 494 4

3 232 8

4 137 16

5 56 32

6 29 64

7 25 128

8 8 2569 6 512

by infinitesimal increments. Now it is highly probable that anyincrease in wealth, no matter how insignificant, will always re-

sult in an increase in utility which is inversely proportionate to

the quantity of goods already possessed.Thus, Daniel Bernoulli argued that if there is an infinitesimal

increase,x, in wealth thenU, the corresponding infinites-imal increase in utility should be inversely proportional to thesize of the fortune already amassed. Then, for some constant a,

U= (a/x)x. (2)Byperspicaciousgeometricalarguments,BernoullideducedthatU(x) = a log x+ b, whereb is a constant. By contrast, mod-ern methods would obtain from (2) the differential equationdU/dx = a/x, which may be integrated in a straightforwardmanner to return the solution given above.

For example, if the utility of an amount of money is measuredby its logarithm then the expected utility arising from playingthe St. Petersburg game is

EP(log Y) =k=1

1

2k log(2k) =

k=1

k

2klog 2 = 2 log 2,

wherePcorresponds to the geometric distribution. This expec-tation corresponds to a monetary amount of exactly four ducats.From this point of view a rational gambler like Paulthat is, aperson who measures marginal utility using logarithmswouldpay any sum less than four ducats to enter the game.

Buffon had a child play the St. Petersburg game 2,048 times.Theresultsofthissimulation,reportedbyJorland(1987,p.168),are reproduced in Table 1.

Based on these empirical frequencies Buffon concluded that,despite the theoretical expected infinite expectation, the St. Pe-tersburg game in practice becomes fair with an entrance fee of

approximately ten ducats.Whitworth (1901) proposed another resolution of the para-dox which is independent of devices for measuring the utility ofmoney. Hissolution considered thepossibility of gamblers ruin,a natural concern for all gamblers with finite resources. Whit-worth assumed that prudent gamblers would place at risk a fixed

percentage, rather than a fixed amount, of their funds, and hedeveloped a procedure for analyzing ventures that involve riskof ruin. It follows from Whitworths study that Pauls entrancefee should depend on the size of his funds.

Feller (1945) proposed a different method to determine en-trance fees which would make the St. Petersburg game fair. Sup-pose Paul chooses to play the game repeatedly. After n games

havebeenplayed,let Rndenotethetotalentrancefeesandlet Sndenote Pauls accumulated receipts. Let us call the game asymp-totically fairif the ratioSn/Rnconverges to1in probabilityasntends to infinity, that is, if for every >0

limn

P

SnRn 1 <

1.

Feller proved that the St. Petersburg game becomes asymptot-ically fair if Rn = n log2 n. From the calculation of Pauls

expected payoff, we see also that the game cannot be asymp-totically fair if there is a fixed and finite entrance fee per game,that is, ifRn = cn where c is a finite constant. However, ifthe entrance fee per game may depend on the number of gamesalready played then, according to Fellers theorem, the St. Pe-tersburg paradox is resolved; see Feller (1969, pp. 235237).

Others have noted that the St. Petersburg game seems unre-alistic, given that natural limits exist on the time and financialresources of Peter and Paul. From the point of view of those crit-ics, the St. Petersburg problem is not even a paradox. As Durand(1957, p. 352) eloquently put it, Peter and Paul are mortal; so,after a misspent youth, a dissipated middle age, and a dissolute

dotage, one of them will die, and the game will ceaseheadsor no heads. Or again, Peters solvency is open to question, forthe stakes advance at an alarming rate . . . Even if Peter and Paulagree to cease after 100 tosses, the stakes, though finite, staggerthe imagination.

In an extensive assessment of the St. Petersburg game,Samuelson (1977) commented: Of course Buffons calculatedratio is nonsensical as a proposed estimate of the mathematicalexpectationof Paul perinfinite Petersburggame.In 2, 048 tosses,Paul may win anything from0to22,048. Why take the result of1 arbitrary sample? And how did Buffon score uncompletedgames? If Buffon correctly applies the method of fair division

between Peter and Paul for games interrupted before their finish,he will find that half the time on the 2, 048th toss the sequenceshowsaTandanincompletegame. . . therefore, Paulsexpectedmoney gain will indeed be infinite . . . Consequently, BuffonsMonte Carlo experiments are childish ways to avoid the trueinfinity in the Petersburg expectations: adultishly understood,they confirm the need to replace Pauls linear utility functionby a sufficiently concave function. At bottom, Samuelson con-cluded, the paradox is no paradox.

Following on the idea of varying entrance fees as initiated byFeller (1945), a deterministic sequence of entrance fees for theSt. Petersburg game,

2 4 2 8 2 4 2 16 2 4 2 8 2 4 2 32 2 4 2. . . ,

was given by Steinhaus(1949). To construct this sequence,placetwos in alternating empty places, then fill every second emptyplace by a four, next fill every second remaining empty spaceby an eight, and so on. Denote by a1, a2, . . .the members ofthis sequence, and let anbe the entrance fee at thenth repeti-tion of a St. Petersburg game. Steinhaus proved that the sampledistribution function ofa1, . . . , anconverges to the distributionfunction ofY asn .

More recently, Csrgo and Simons (1993) provided an ex-tensive discussion of Steinhaus sequence of entrance fees, andthere now exists an extensive literature on asymptotic theory for

The American Statistician, August 2004, Vol. 58, No. 3 227


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St. Petersburg games. We refer to Adler (1990), Berkes, Cski,and Csrgo (1999), Csrgo and Simons (1996), and Csrgo(2003) for recent developments in this area and for further refer-ences. Especially intriguing is the remarkable article by Csrgoand Simons (2002) which reveals new paradoxes within thestructure of the classical St. Petersburg paradox; in particular,their results show that, paradoxically, there may be a very dif-ferent outcome ifndistinct Pauls play one St. Petersburg gameeach than if one Paul playsngames.

4. RELATED PARADOXES AND GAMES

Daniel Bernoullis response to the paradox has not been uni-versally regarded as satisfactory (see Weirich 1984), this beingdue to a controversy regarding the relationship between addi-tional amounts of ducats and the measurement of utility. More-over, even if it is assumed that additional sums of money havedecreasing marginal utilities and also that it is appropriate tocalculate utilities by taking logarithms, the problem is not en-tirely resolved. Indeed, Samuelson (1977) discussed at lengththe Super-Petersburg paradoxes constructed by K. Menger;here, for a given unbounded utility function, the payoffs growsufficiently fast that the resulting expected utility is infinite. Forinstance, in a St. Petersburg game with logarithmic utility inwhich Paul receives22

k

ducats if the first head is obtained at thekth toss, Pauls expected payoff again is infinite, andthe paradoxreturns. If a new, unbounded utility function were to be devisedfor this game so as to return a finite expectation, then a new setof more rapidly increasing payoffs can be designed so that theresulting expectation again is infinite. By repeatedly adjustingthe payoff strategy, it is seen that the measurement of utility bylogarithms can always be made to result in an infinite expec-tation, and it becomes clear that the paradox can be avoided ifand only if the utility function is bounded, an observation due

to Menger.Despite objections to the calculation of utilities by the use

of logarithms, there are instances in which this approach is en-tirely natural and has significant advantages over other measuresof utility. For one such example within the context of optimalgambling systems, we refer the reader to Thorp (1969, p. 289).

There arenumerous paradoxes analogousto theSt.Petersburgproblem. One of them is the following paradox, communicatedto Szkely by Guttman in 1983: Peter entraps Paul into tossinga fair coin repeatedly. Whenk tosses are needed to observe thefirsthead,Paulreceives (2)k ducatsfromPeter.Inotherwords,ifk is even then Paul receives2k ducats from Peter, but ifk is

odd then Paul pays2k

ducats to Peter. Should Paul be happy orunhappy about having been entrapped into playing this game?The answer is that Paul should both be happy and unhappy.

To explain this paradoxical conclusion, we partition the set ofpositiveintegers into subsets {1, 2, 4}, {3, 6, 8}, {5, 10, 12},andsoon,sothatthe rthsubsetinthispartitionis {2r1, 4r2, 4r}.Then the gamecan be viewed asconsisting of a collection of sub-games, each corresponding to one of the subsets in our partition.With probability 2(2r1)+ 2(4r2)+ 24r, Paul has receivedthe subgame which awards(2)k ducats with probability

2k

2(2r1)

+ 2(4r2)

+ 24r

, k= 2r 1, 4r 2, 4r.

It is straightforward to calculate that each of these subgames isfavorable to Paul, that is, Paul has positive expected payoff fromeach subgame. Therefore, Paul should be happy.

On the other hand, we can also partition the set of positiveintegers into subsets {1, 2, 3}, {4, 5, 7}, {5, 10, 12}, and so on,in which therth subset in this partition is {2r, 4r 3, 4r 1}.With probability22r + 2(4r3) + 2(4r1), Paul receives thesubgame which awards(2)k ducats with probability

2k

22r + 2(4r3) + 2(4r1), k= 2r, 4r 3, 4r 1.

Each of these subgames is unfavorable to Paul, so he also shouldbe unhappy.

Let us restate Guttmans problem in terms of conditional ex-pectations. IfXandYare jointly distributed random variables,we denote byP(dx|Y)the conditional probability distributionofXgivenY. Then the conditional expectation ofXgivenYis

xP(dx|Y).

Surprisingly, there exist randomvariables X, Y,and Zsuch thatE(X|Y) > 0 > E(X|Z)with probability one. For example,let Xdenote the overall reward to Paul in Guttmans game.Let Y = r if Paul receives the rth favorable game in whichX {2r 1, 4r 2, 4r}. Also, let Z = r if Paul receivestherth unfavorable game in which X {2r, 4r 3, 4r 1}.Then it is simple to verify that E(X|Y) > 0 > E(X|Z)withprobability one.

Stories of St. Petersburg games date to antiquity. There is theclassic, perhaps apocryphal, story of the king who decided toreward an old and faithful peasant with a generous sum of gold.Instead, the peasant gently requested his king to give him one

grain of corn on the first day of the month, two grains of cornon the second day, four grains on the third day, etc. The kingwas so pleased by his servants deep modesty that he happilygranted the request, and his kingdom was bankrupt before themonth ended.

An interesting example of a St. Petersburg offer involved KenHarrelson, a Major League baseball player with the ClevelandIndians from 19691971. As reported by Smith (1988, p. 157),Harrelson proposed to Gabe Paul, general manager of the Cleve-land Indians, a St. Petersburg deal in which Harrelson wouldplay the entire baseball season without salary except for 50cents doubled for every home run he hit. Harrelsons salary

would then be 50 cents if he hit only one home run, one dollarif he hit two home runs, two dollars for three home runs, andso on. Gabe Paul was wise enough to decline Harrelsons offer,especially since Harrelson went on to hit 30 home runs duringthat season!

Currently, a popular television game show, Who Wants to be aMillionaire?, offers players the chance to win up to one milliondollars. Players are offered a sequence of questions, each withfour possible answers. If players answer a question correctlythen they can walk away with their prize, or they can chooseto answer a new question, thereby taking the risk of losing allif they answer incorrectly or doubling their prize by answeringcorrectly.

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Many examples of St. Petersburg games and their generaliza-tions are studied in the statistics, economics, and mathematicsliterature. A search of databases covering the literature in thosefields will recover dozensof articles studying proposed solutionsand applications.

5. THE VALUATION OF GROWTH STOCKS

A remarkable application of the St. Petersburg paradox, dueto Durand (1957), relates to the valuation of the common stocks

of growth companies. Here, a growthcompany is onewhoserevenues are growing significantly faster than the overall econ-omy; and the stocks or shares of these companies, or even thecompanies themselves, are commonly referred to as growthstocks. We shall review this application in detail because ofits applicability to financial events in the late 1990s and early2000s.

As background, we recall that by early 2000, unprecedentedincreaseshadoccurred over the previous three years in the pricesof growth stocks. These increases in stock prices sparked alively debate over whether investors were wise to purchase theseshares, or foolish not to purchase greater amounts of shares be-fore prices increased even further.

On the one hand were skeptics, including Alan Greenspan,chairman of the board of governors of the United States FederalReserve System, who raised concerns about inflationary pres-sures resulting from increased stock prices. In a now-famousspeech on December 5, 1996, Greenspans question, But howdo we know when irrational exuberance has unduly escalatedasset values, which then become subject to unexpected and pro-longedcontractionsas they have in Japan over the past decade?,was followed by dramatic fluctuations on world financial mar-kets.

On the other hand were believers, including many mutualfunds which had purchased substantial numbers of shares of

high-tech companies. Indeed, the Wall Street Journal reportedonNovember 19,1999, that 59 mutual funds hadamassed increasesof more than 100% during the period January 1November17, 1999. One fund, the Nicholas-Applegate Global Technol-ogy Fund, which specialized in high-tech stocks, had increasedin price during the same period by the astounding amount of325%, or about 1% per day, an amount which the Journalcom-mented, tongue-in-cheek, as possibly insufficient for patientinvestors.

Following Durand (1957), we consider a modified St. Peters-burg game in which Peter is a growth company and Paul is aprospective purchaser of Petersstock. We assume that the prob-

ability of tossing heads is i/(1 + i),i >0; then the probabilityof tails is1/(1 + i).Next, suppose that the corresponding payoffs are a series of

increasing payments in which Peter pays Paul D ducats if thefirst toss results in a tail,D(1 + g)ducats if the second toss isa tail,D(1 + g)2 ducats if the third toss is a tail, and so on, andthis continues until the toss results in heads, at which point thegame ends. Ifk tosses are needed for the game to end then thetotal payment to Paul is

k2j=0

D(1 + g)j =D[(1 + g)k1 1]

g . (3)

Because heads and tails appear with probabilitiesi/(1 + i)and1/(1 + i), respectively, then the payment (3) occurs with prob-abilityi/(1 + i)k. As Durand (1957) observed, Pauls expectedpayoff is given by the double summation

k=1

i

(1 + i)k

k2j=0

D(1 + g)j .

This double sum is evaluated by substituting for the inner sum

the closed form expression (3). Alternatively, by reversing theorder of summation and evaluating the resulting inner sum, wefind that Pauls expected payoff is

k=1

D(1 + g)k1

(1 + i)k =

D/(i g), if g < i,

, if g i. (4)

In summary, ifg < i, then Pauls expected payoff isD/(i g).On the other hand, ifg i, then the infinite series diverges andPauls expected payoff is infinite, in which case a realistic Paulmay wisely decline to pay the corresponding entrance fee.

In the context of appraising the values of financial securities,

the parameteri represents a compound interest rate (or an ef-fective rate of interest); equivalently, the present value of a loanof one dollar to be repaid one year in the future is 1/(1 + i). Inthe appraisal of fair value for a companys shares, g representsthe growth rate of the company as measured by the compoundincrease in revenue per share.

A widely accepted approach to estimating a fair value forPeters stock is to discount all future dividends in perpetuity.Here, a fair value for Peters stock is estimated by the presentvalueof all future dividends. Denoteby EnPeters earnings (i.e.,profits) per share in year n. Also, let Bn denote Peters bookvalue, or net asset value, per share in the same year. Further,

denote byDnthe total of Peters paid-out dividends per sharein yearn. A moments reflection makes it clear that changes inbook valuefromyear-to-year are equal to the difference betweenearnings and dividends paid, that is, for alln 1

Bn+1 Bn= En Dn.

InestimatingfairvalueforPetersstock,itiscommonpracticefor Paul to assume that the ratiosr = En/Bnand p = Dn/Enare independent ofn. This assumption implies that Bn+1 Bn,the change in book value from year n to year n + 1, is a constantmultiple ofEn:

Bn+1 Bn = En Dn= (1 p)En= (1 p)r Bn.

Therefore Peters dividends, book value, and earnings all aregrowing at a constant rate, g = (1 p)r. In this context the sum(4) represents aperpetualseries of dividend payments, startingat D ducats, growing at a constant rate g, and discounted atrate i in perpetuity. If i > g, then the sum (4) converges toD1/(i g) =p E1/(i g), which represents an estimate of fairvalue for one share of Peters stock.

Ifi g, then the series (4) diverges, and we now have a formof the St. Petersburg paradox in which the practice of discount-ing future dividends at a uniform rate in perpetuity leads to aparadoxical result.



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6. CONCLUDING REMARKS

The St. Petersburg paradox explains some of the unprece-dented increases in the prices of high-tech growth stocks in thelate 1990s. During that period, the Federal Reserve Systemsdiscount rate was near a historical low; in the context of Du-rands article, i was very small. Moreover, purchasers of growthstocks assumed that g, the growth rate of a typical high-techcompany, would remain high in perpetuity. The outcome wasthati < g; indeed, even more extreme was that for many high-

techcompanies, i/g 0.Bydiscountingearningsanddividendsin perpetuity, any use of the valuation formula (4) led to esti-mated valuations of tech-stock share prices at levels as high asthe appraiser cared to make them.

Having applied the valuation formula (4) to obtain exorbitantestimated valuations for many high-tech growth stocks, stockpurchasers bought avidly, thereby forcing prices to extreme lev-els. By late 2000, stock prices underwent the prolonged con-tractions predicted by Greenspan, with subsequent unprece-dented losses to corporate and individual stock buyers. Threeyears later, many formerly avidly sought-after high-tech com-panies and mutual funds were defunct.

Purchasers of high-tech growth stocks in the late 1990s over-lookedcompletely thecautionary analyses of Durand(1957)andMalkiel (1963) regarding the perpetual discounting of earningsor dividends; even more regrettable was the propensity to ignorethe sage advice of Graham (1985).It is striking that Durands ar-ticlewascompletelyoverlookedinthe1990sdespitethefactthatit was written during a similar period in the 1950s when stockpurchasers had been eagerly purchasing the high-tech stocks ofthe time, and despite the clear observation by Durand and ear-lier authors that the divergence of the series (4) when i < gwould lead to infinite estimated fair prices for common stocks.As a clear warning, Durand (1957, p. 349), quoting Clendeninand Van Cleave (1954), commented that he had not yet seenany growth stocks marketed at the price of infinity dollars pershare, but [would] hereafter be watching. Unfortunately, thesecomments were overlooked or were unpersuasive to high-techstock buyers of the 1950s and they eventually came to grief, asdid similar buyers in the 1990s.

We close with two comments from Mark Twain who noted,for the benefit of stock market speculators, that

There are two times in a mans life when he should not speculate: when he cantafford it, and when he can.

Also, from TwainsPuddnhead Wilsons Calendarcomes the

quotation,

October. This is one of thepeculiarlydangerous months to speculatein stocksin.The others are July, January, September, April, November, May, March, June,December, August and February.

[Received October 2003. Revised March 2004.]

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Bernoulli,D. (1738), Specimen Theoriae Novae de Mensura Sortis, Commen-tarii Academiae Scientiarum Imperialis Petropolitanea, V, 175192. (Trans-lated and republished as Exposition of a New Theory on the Measurementof Risk, (1954),Econometrica, 22, 2336.)

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Clendenin, J. C., and Van Cleave, M. (1954), Growth and Common StockValues, The Journal of Finance, 9, 365376.

Csrgo, S. (2003), Merge Rates for Sums of Large Winnings in GeneralizedSt. Petersburg Games,Acta Scientiarum Mathematicarum(Szeged),69,441454.

Csrgo, S., and Simons, G. (19931994), On Steinhaus Resolution of the St.Petersburg Paradox,Probability and Mathematical Statistics, 14, 157172.

(1996), A Strong Law of Large Numbers for Trimmed Sums, withApplications to Generalized St. Petersburg Games,Statistics & Probability

Letters, 26, 6573.

(2002), The Two-Paul Paradox and the Comparison of Infinite Expec-tations, inLimit Theorems in Probability and Statistics, eds. I. Berkes et al.,Budapest: Jnos Bolyai Mathematical Society, pp. 427455.

DeGroot,M.H.(1970), Optimal Statistical Decisions, NewYork: McGraw-Hill.

Dehling, H. G. (1997), Daniel Bernoulliand theSt. Petersburg Paradox,NieuwArchief voor Wiskunde, 15, 223227.

Durand, D. (1957), Growth Stocks and the Petersburg Paradox, The Journalof Finance, 12, 348363.

Dutka, J. (1988), On the St. Petersburg Paradox,Archive for History of ExactSciences, 39, 1339.

Feller, W. (1945), Note on the Law of Large Numbers and Fair Games,Annals of Mathematical Statistics, 16, 301304.

Graham, B. (1985), The Intelligent Investor: A Book of Practical Counsel (4thed.), New York: Harper & Row.

Jorland, G. (1987), The Saint Petersburg Paradox 17131937, in The Prob-abilistic Revolution (vol. 1), eds. L. Kreger, et al., Cambridge, MA: MITPress, pp. 157190.

Malkiel, B. (1963), Equity Yields, Growth, and the Structure of Share Prices,American Economic Review, 53, 10041031.

de Montmort, P. R. (1713), Essay dAnalyse sur les Jeux de Hazard(2nd ed.)

Paris.Samuelson, P. A. (1977), St. Petersburg Paradoxes: Defanged, Dissected, and

Historically Described,Journal of Economic Literature, 15, 2455.

Smith, G. (1988),Statistical Reasoning(2nd ed.), Boston, MA: Allyn & Bacon.

Steinhaus, H. (1949), The So-Called Petersburg Paradox,Colloquium Math-ematicum, 2, 5658.

Thorp, E. O. (1969), Optimal Gambling Systems for Favorable Games, Re-views of the International Statistical Institute, 37, 273293.

Weirich, P. (1984), TheSt. Petersburg Gamble and Risk,Theory and Decision,17, 193202.

Whitworth, W. A. (1901),Choice and Chance(5th ed.), Cambridge: Deighton,Bell, & Co.

ADDITIONAL READING

The following references provide additional information onasymptotic approaches to resolving the paradox and the methodof variable entrance fees.

Csrgo, S. (2002), Rates of Merge in Generalized St. Petersburg Games,ActaScientiarum Mathematicarum (Szeged), 68, 815847.

Csrgo, S., and Dodunekova, R. (1991), Limit Theorems for the PetersburgGame,in Sums, TrimmedSumsand Extremes,eds.M.G.Hahnetal.,Boston:Birkhuser, pp. 285315.

Feller, W. (1969), An Introduction to Probability Theory and Its Applications(vol. I, 2nd ed.), New York: Wiley.

Martin-Lf,A. (1985), A Limit Theorem Which Clarifies the Petersburg Para-dox,Journal of Applied Probability, 22, 634643.

230 History Corner
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The following references provide additional information onthe St. Petersburg paradox and its connections with the conceptof marginal utility.

Arntzenius, F., and McCarthy, D. (1997), The Two Envelope Paradox and In-finite Expectations,Analysis, 57, 4250.

Machina, M. J. (1982), Expected Utility Analysis Without the IndependenceAxiom,Econometrica, 50, 277323.

Shafer, G. (1988), The St. Petersburg Paradox, in Encyclopedia of StatisticalSciences, Volume 8, eds. N. L. Johnson and S. Kotz, New York: Wiley, pp.865870.

Sinn, H.-W. (1989),Economic Decisions Under Uncertainty(2nd ed.), Heidel-

berg: Physica-Verlag.

The following reference provides additional information onthe St. Petersburg paradox and its application to the valuation ofgrowth stocks.

Bernstein, P. L. (1996),Against the Gods: The Remarkable Story of Risk, NewYork: Wiley.

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St Petersburg Paradox and Tech Stocks 2000