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[World Scientific Handbook in Financial Economics Series] The Kelly Capital Growth Investment...
[World Scientific Handbook in Financial Economics Series] The Kelly Capital Growth Investment Criterion Volume 3 (Theory and Practice)  The Kelly Criterion in Blackjack Sports Betting, and the Stock Market
MacLean, Leonard C, Thorp, Edward O, Ziemba, William Tكم أعجبك هذا الكتاب؟
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المجلد:
10.1142/75
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2011
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english
DOI:
10.1142/9789814293501_0054
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789 Chapter 9 54 THE KELLY CRITERION IN BLACKJACK SPORTS BETTING, AND THE STOCK MARKET* EDWARD O. THORP Edward O. Thorp and Associates, Newport Beach, CA 92660, USA The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. Contents Abstract Keywords 1. Introduction 2. Coin tossing 3. Optimal growth: Kelly criterion formulas for practitioners 386 386 387 388 3.1. The probability ofreaching a fixed goal on or before 11 trials 392 394 3.2. The probability of ever being reduced to a fraction x of this initial bankroll 392 3.3. The probability of being at or above a specified value at the end of a specified number of trials 3.4. Continuous approximation of expected time to reach a goal 395 396 3.5. Comparing fixed fraction strategies: the probability that one strategy leads another after 11 trials 4. 5. 6. 7. The long run: when will the Kelly strategy "dominate"? Blackjack Sports betting Wall street: the biggest game 7.1. Continuous approximation 7.2. The (almost) real world 7.3. The case for "fracti onal Kelly" 7.4. A remarkable formula 8. A case study 8.1. The constraints 8.2. The analysis and results 8.3. The recommendation and the result 8.4. The theory for a portfolio of securities 396 398 399 401 405 406 409 411 414 415 416 416 417 418 Paper presented at: The IOth International Conference on Gambling and Risk Taking. Montreal, June 1997. published in: Finding the Edge: Mathematical Analysis of Casino Games. edited by O. Vancura. 1.A. Cornelius, w.R. Eadington. 2000. Corrections added April 20, 2005. Handbook of Asset and Liability Management, Volume I Edited by S.A. Zenios and W. T. Ziemha © 2006 Published by Elsevier B. V. DOl: JO.JOJ6/SJ8720978(06)OJ009X 790 E. 0. Thorp 386 £.0. Thorp The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. 9. My experience with the Kelly approach 10. Conclus; ion Acknow ledgements Appendix A. Integrals for deriving moments of E 00 Appendix B. Derivation of formula (3.1) Appendix C. Expected time to reach goal References 419 420 420 420 421 423 428 Abstract The central problem for gamblers is to find positive expectation bets. But the gambler also needs to know how to manage his money, i.e., how much to bet. In the stock market (more inclusively, the securities markets) the problem is similar but more complex. The gambler, who is now an "investor", looks for "excess risk adjusted return". In both these settings, we explore the use of the Kelly criterion, which is to maximize the expected value of the logarithm of wealth ("maximize expected logarithmic utility"). The criterion is known to economists and financial theorists by names such as the "geometric mean maximizing portfolio strategy", maximizing logarithmic utility, the growthoptimal strategy, the capital growth criterion, etc. The author initiated the practical application of the Kelly criterion by using it for card counting in blackjack. We will present some useful formulas and methods to answer various natural questions about it that arise in blackjack and other gambling games. Then we illustrate its recent use in a successful casino sports betting system. Finally, we discuss its application to the securities markets where it has helped the author to make a thirty year total of 80 billion dollars worth of "bets". Keywords Kelly criterion, betting, long run investing, portfolio allocation, logarithmic utility, capital growth JEL classification: C61, D81, G 1 The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 791 Ch. 9: 387 The Kelly Criterion in Blackjack Sports Betting, and the Stock Market The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. 1. Introduction The fundamental problem in gambling is to find posItIve expectation betting opportunities. The analogous problem in investing is to find investments with excess riskadjusted expected rates of return. Once these favorable opportunities have been identified, the gambler or investor must decide how much of his capital to bet. This is the problem which we consider here. It has been of interest at least since the eighteenth century discussion of the St. Petersburg Paradox (Feller, 1966) by Daniel Bernoulli. One approach is to choose a goal, such as to minimize the probability of total loss within a specified number of trials, N. Another example would be to maximize the probability of reaching a fixed goal on or before N trials (Browne, 1996). A different approach, much studied by economists and others, is to value money using a utility function. These are typically defined for all nonnegative real numbers, have extended real number values, and are nondecreasing (more money is at least as good as less money). Some examples are U (x) = x{/ , 0 :::;; a < 00 , and U (x) = log x, where log means loge ' and log 0 = 00. Once a utility function is specified, the object is to maximize the expected value of the utility of wealth. Daniel Bernoulli used the utility function logx to "solve" the St. Petersburg Paradox. (But his solution does not eliminate the paradox because every utility function which is unbounded above, including log, has a modified version of the St. Petersburg Paradox.) The utility function log x was revisited by Kelly (1956) where he showed that it had some remarkable properties. These were elaborated and generalized in an important paper by Breiman (1961). Markowitz (1959) illustrates the application to securities. For a discussion of the Kelly criterion (the "geometric mean criterion") from a finance point of view, see McEnally (1986). He also includes additional history and references. I was introduced to the Kelly paper by Claude Shannon at M.l.T. in 1960, shortly after I had created the mathematical theory of card counting at casino blackjack. Kelly's criterion was a bet on each trial so as to maximize E log X , the expected value of the logarithm of the (random variable) capital X. I used it in actual play and introduced it to the gambling community in the first edition of Beat the Dealer (Thorp, 1962). If all blackjack bets paid even money, had positive expectation and were independent, the resulting Kelly betting recipe when playing one hand at a time would be extremely simple: bet a fraction of your current capital equal to your expectation. This is modified somewhat in practice (generally down) to allow for having to make some negative expectation "waiting bets", for the higher variance due to the occurrence of payoffs greater than one to one, and when more than one hand is played at a time. Here are the properties that made the Kelly criterion so appealing. For ease of understanding, we illustrate using the simplest case, coin tossing, but the concepts and conclusions generalize greatly. 792 E. 0. Thorp 388 E.O. Thorp The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. 2. Coin tossing Imagine that we are faced with an infinitely wealthy opponent who will wager even money bets made on repeated independent trials of a biased coin. Further, suppose that on each trial our win probability is p > 1/ 2 and the probability oflosing is q = 1  p. Our initial capital is Xo. Suppose we choose the goal of maximizing the expected value E (X,,) after n tria~s. How much should we bet, Bk, on the kth trial? Letting Tk = 1 if the kth trial is a win and Tk = I ifit is a loss, then Xk = X k I +TkBk for k = 1, 2, 3, ... , and X" = Xo + :L%=I TkBk. Then " " Since the game has a positive expectation, i.e., p  q > 0 in this even payoff situation, then in order to maximize E(X,,) we would want to maximize E(Bd at each trial. Thus, to maximize expected gain we should bet all of our resources at each trial. Thus BI = Xo and if we win the first bet, B2 = 2X 0, etc. However, the probability of ruin is given by 1  p" and with p < I, lim,, +oo [ I  p"] = I so ruin is almost sure. Thus the "bold" criterion of betting to maximize expected gain is usually undesirable. Likewise, if we play to minimize the probability of eventual ruin (i .e., "ruin" occurs if Xk = 0 on the kth outcome) the wellknown gambler's ruin formula in Feller (\966) shows that we minimize ruin by making a minimum bet on each trial, but this unfortunately also minimizes the expected gain . Thus "timid" betting is also unattractive. This suggests ·an intermediate strategy which is somewhere between maximizing E(X,,) (and assuring ruin) and minimizing the probability of ruin (and minimizing E (X,,». An asymptotically optimal strategy was first proposed by Kelly (\956). In the cointossing game just described, since the probabilities and payoffs for each bet are the same, it seems plausible that an "optimal" strategy will involve always wagering the same fraction f of your bankroll. To make this possible we shall assume from here on that capital is infinitely divisible. This assumption usually does not matter much in the interesting practical applications. If we bet according to Bi = f Xi  I, where 0 ~ f ~ I, this is sometimes called "fixed fraction" betting. Where Sand F are the number of successes and failures, respectively, in n trials, then our capital after n trials is X" = Xo(l + (1where S + F = n. With f in the interval 0 < f < I, Pr(X" = 0) = O. Thus "ruin" in the technical sense of the gambler's ruin problem cannot occur. "Ruin" shall henceforth be reinterpreted to mean that for arbitrarily small positive E, lim" +oo [Pr(X,, ~ E)] = I. Even in this sense, as we shall see, ruin can occur under certain circumstances. We note that since n S nF, e"log [ X ] 1/ " X~ The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 793 Ch. 9: 389 The Kelly Criterion .in) Blackjack Sporls Betting, and the Stock Market the quantity X"]! /,, G,,(f) = log [ Xo S n = log(1 F + j) + Iog(l n j) measures the exponential rate of increase per trial. Kelly chose to maximize the expected value of the growth rate coefficient, g(f), where The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. g(f) = E{ 10g[~~r/"}. = E{ ~ 10g(1 + j) + = P 10g(1 + j) + q 10g(1  j). =l~g(1  j)} Note that g(f) = (1 /n)E(logX,,)  (l/n)logXo so for n fixed, maximizing g(f) is the same as maximizing E log X". We usually will talk about maximizing g(f) in the discussion below. Note that , g (f) p = 1+ f  q 1 f = (1 pqf + j) (1 _ j) =0 when f = f* = p  q. Now g"(j) = p/(1 + j)2  q/(1  j)2 < 0 so that g' (f) is monotone strictly decreasing on [0, 1). Also g' (0) = p  q > 0 and lim/> ! g' (f) = 00. Therefore by the continuity of g' (j), g(f) has a unique maxie mum at f = f*, where g(f*) = p log p + q log q + log 2 > O. Moreover, g(O) = 0 and lim j>q g(f) = 00 so there is a unique number f, . > 0, where 0 < f* < fe < 1, such that g(fc) = O. The nature of the function g(f) is now apparent and a graph of g(f) versus f appears as shown in Figure I . The following theorem recounts the important advantages of maximizing g(f). The details are omitted here but proofs of (i)(iii), and (vi) for the simple binomial case can be found in Thorp (1969); more general proofs of these and of (iv) and (v) are in Breiman (1961). Theorem 1. (i) If g(f) > 0, then limn > oo X" = 00 almost surely, i.e., for each M, Pr[liminfn >oo XII > M] = 1; (ii) If g(j) < 0, then limll >oo X" = 0 almost surely; i.e., for each 8 > 0, Pr[limsuPn> oo X" < 8] = 1; (iii) If g(f) = 0, then lim sUPn> 00 Xn = 00 a.s. and lim inf,, > 00 Xn = 0 a.s. (iv) Given a strategy cJ>* which maximizes E log X" and any other "essentially different" strategy cJ> (not necessarily a fixed fractional betting strategy), then lim'l>oo Xn(cJ>*)/ X,,(cJ» = 00 a.s. (v) The expected time for the current capital X" to reach any fixed preassigned goal C is, asymptotically, least with a strategy which maximizes E log X". (vi) Suppose the return on one unit bet on the ith trial is the binomial random variable Ui; further, suppose that the probability of success is Pi, where 1/2 < Pi < l. 794 E. 0. Thorp 390 E.O. Thorp The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. G(f) o f Fig. I. Then E log XII is maximized by choosing on each trial the fraction maximizes E log(1 + if Vi). 1;* = Pi  qi which Part (i) shows that, except for a finite number of terms, the player's fortune XII will exceed any fixed bound M when f is chosen in the interval (0, fe) . But, if f > f e, part (ii) shows that ruin is almost sure. Part (iii) demonstrates that if f = fe, X I1 will (almost surely) oscillate randomly between and +00. Thus, one author's statement that X I1 + Xo as n + 00, when f = f e, is clearly contradicted. Parts (iv) and (v) show that the Kelly strategy of maximizing E log XII is asymptotically optimal by two important criteria. An "essentially different" strategy is one such that the difference E In X,:  E In XI1 between the Kelly strategy and the other strategy grows faster than the standard deviation of In X~  In XII' ensuring P (In X~  In XII > 0) + 1. Part (vi) establishes the validity of utilizing the Kelly method of choosing 1;* on each trial (even if the probabilities change from one trial to the next) in order to maximize E log X n . ° Example 2.1. Player A plays against an infinitely wealthy adversary. Player A wins even money on successive independent flips of a biased coin with a win probability of p = .53 (no ties). Player A has an initial capital of Xo and capital is infinitely divisible. The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 795 Ch. 9: 391 The Kelly Criterion in Blackjack Sports Betting, and the Stock Market Applying Theorem I(vi), f* = p  q = .53  .47 = .06. Thus 6% of current capital should be wagered on each play in order to cause XII to grow at the fastest rate possible consistent with zero probability of ever going broke. If Player A continually bets a fraction smaller than 6%, XII will also grow to infinity but the rate will be slower. If Player A repeatedly bets a fraction larger than 6%, up to the value f e, the same thing applies. Solving the equation g(f) = .5310g(l + f) + .4710g(1  f) = 0 numerically on a computer yields fe = .11973. So, if the fraction wagered is more than about 12%, then even though Player A may temporarily experience the pleasure of a faster win rate, eventual downward fluctuations will inexorably drive the values of XII toward zero. Calculation yields a growth coefficient of g(f*) = f(.06) = .001801 so that after n successive bets the log of Player A's average bankroll will tend to .001801n times as much money as he started with. Setting .00180 I n = log 2 gives an expected time of about n = 385 to double the bankroll. The Kelly criterion can easily be extended to uneven payoff games. Suppose Player A wins b units for every unit wager. Further, suppose that on each trial the win probability is p > 0 and pb  q > 0 so the game is advantageous to Player A. Methods similar to those already described can be used to maximize g(f) = E 10g(XIII Xo) = P log(l + bf) + q log(l  f). Arguments using calculus yield f* = (bp  q) I b, the optimal fraction of current capital which should be wagered on each play in order to maximize the growth coefficient g(f). This formula for f* appeared in Thorp (1984) and was the subject of an April 1997 discussion on the Internet at Stanford Wong's website, http://bj2I.com (miscellaneous free pages section). One claim was that one 'can only lose the amount bet so there was no reason to consider the (simple) generalization of this formula to the situation where a unit wager wins b with probability p > 0 and loses a with probability q . Then if the expectation m == bp  aq > 0, f* > 0 and f* = mlab. The generalization does stand up to the objection. One can buy on credit in the financial markets and lose much more than the amount bet. Consider buying commodity futures or selling short a security (where the loss is potentially unlimited). See, e.g., Thorp and Kassouf (1967) for an account of the E.L. Bruce short squeeze. For purists who insist that these payoffs are not binary, consider selling short a binary digital option. These options are described in Browne (1996). A criticism sometimes applied to the Kelly strategy is that capital is not, in fact, infinitely divisible. In the real world, bets are multiples of a minimum unit, such as $1 or $.0 I (penny "slots"). In the securities markets, with computerized records, the minimum unit can be as small as desired. With a minimum allowed bet, "ruin" in the standard sense is always possible. It is not difficult to show, however (see Thorp and Walden, 1966) that if the minimum bet allowed is small relative to the gambler's initial capital, then the probability of ruin in the standard sense is "negligible" and also that the theory herein described is a useful approximation. This section follows Rotando and Thorp (1992). 796 E. O. Thorp 392 £.0. Thorp The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. 3. Optimal growth: Kelly criterion formulas for practitioners Since the Kelly criterion asymptotically maximizes the expected growth rate of wealth, it is often called the optimal growth strategy. It is interesting to compare it with the other fixed fraction strategies. I will present some results that I have found useful in practice. My object is to do so in a way that is simple and easily understood. These results have come mostly from sitting and thinking about "interesting questions". I have not made a thorough literature search but I know that some of these results have been previously published and in greater mathematical generality. See, e.g., Browne (1996, 1997) and the references therein. 3. J. The probability of reaching a fixed goal on or before n trials We first assume coin tossing. We begin by noting a related result for standard Brownian motion. Howard Tucker showed me this in 1974 and it is probably the most useful single fact I know for dealing with diverse problems in gambling and in the theory of financial derivatives. For standard Brownian motion X (t), we have p(sup[X(t)  (at +b)] ~ 0, 0 ( t ( T) + e 2{/b N(a  (3) (3.1) where a = ag and f3 = b/ g. See Figure 2. See Appendix B for Tucker's derivation = N(a  (3) of (3.1). In our application a < O,b > o so we expect limT ...... oo P(X(t) ~ at+b, 0 ( t ( T) =1. Let f be the fraction bet. Assume independent identically distributed (i .d.d.) trials Yi , i = I, ... ,n, with P(Yi = I) = p > 1/2, P(Yi = I) = q < 1/2; also assume p < 1 to avoid the trivial case p = I. X(!) /' (0, b) / slope a< ° /' /' at+ b T (0,0) b ! ( IOT ,O) Fig. 2. The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 797 Ch. 9: 393 The Kelly Criterion in Blackjack Sports Belling, and the Stock Market Bet a fixed fraction f, 0 < f < 1, at each trial. Let Vk be the value of the gambler or investor's bankroll after k trials, with initial value Vo . Choose initial stake Vo = 1 (without loss of generality); number of trials n; goal C > I. What is the probability that Vk ?': C for some k, 1 :s; k :s; n? This is the same as the probability that log Vk ?': log C for some k , I :s; k :s; n. Letting In = loge we have: k Vk = n(l + Yd) and The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. i I k In Vk = Lln(l + Yif) , i=1 k ElnVk = LEln(l +Yif), i=l k Var(ln Vk) = L Var(ln(l + Yi f)), i=1 + Yi f) = p In(l + f) + q In(l  f) == m == g(f), Var[ln(1 + Yif)] = p[ln(1 + f)]2 + q[ln(1  f)]2  m 2 = (p _ p2)[ln(1 + f)]2 + (q _ q2)[ln(l _ f)]2  2pq In(l + f) In(l  f) = pq ([In ( 1 + f)]2  21n( 1 + f) In(l  f) + [InC 1 = pq{ln[(1 + f)/(I  f)]}2 == s2. E In(l f)]2} Drift in n trials: mn . Variance in n trials: s2n. In Vk ?': InC, 1 :s; k:S; n, iff k L In(l + Yi f) ?': In C, I :s; k :s; n, iff i=1 k Sk == L[ln(l + fif)  m] ?': InC  mk, I :S; k :s; n, i=l We want Prob(Sk ?': In C  mk , I :s; k :s; n). Now we use our Brownian motion formula to approximate SII by Prob(X (t) ?': In Cmt/s 2 , 1 :s; t :s; s2n) where each term of SII is approximated by an XU), drift 0 and 798 E. 0. Thorp 394 E.O. Thorp variance s2 (0 :::::; t :::::; s2, s2 :::::; t :::::; 2s2 , .. . , (n  l)s2 :::::; t :::::; ns 2). Note: the approximation is only "good" for "large" n. Then in the original formula (3.1): T = s2n , The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. b = InC , a = (X = afl = m..jii/s, m/s2, f3 = b/fl = InC/s..jii. Example 3.1. C =2, n = 104 , q = .51, = .49, f = .0117 , p m = .000165561, 05 2 = .000136848. Then PO = .9142. Example 3.2. Repeat with f = .02, then m = .000200013 , 05 2 = .000399947 and PO = .9214. 3.2. The probability of ever being reduced to a fraction x of this initial bankroll This is a question that is of great concern to gamblers and investors. It is readily answered, approximately, by our previous methods. Using the notation of the previous section, we want P(Vk :::::; x for some k , 1 :::::; k :::::; 00). Similar methods yield the (much simpler) continuous approximation formula: Prob(·) = e 2"h where a = m/s2 and b = Inx which can be rewritten as Probe) = x ;\ (2m/o5 2 ) where /\ means exponentiation. (3 .2) The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 799 Ch. 9: 395 The Kelly Criterion in Blackjack Sports Betting. and the Stock Market Example 3.3. p = .51, f = 1* = .02, 2m/s = 1.0002, The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. Probe) ":,, x. We will see in Section 7 that for the limiting continuous approximation and the Kelly optimal fraction 1*, P (VI; (f*) ~ x for some k ) 1) = x. My experience has been that most cautious gamblers or investors who use Kelly find the frequency of substantial bankroll reduction to be uncomfortably large. We can see why now. To reduce this, they tend to prefer somewhat less than the full betting fraction f*. This also offers a margin of safety in case the betting situations are less favorable than believed. The penalty in reduced growth rate is not severe for moderate underbetting. We discuss this further in Section 7. 3.3. The probability of being at or above a specified value at the end of a specified number of trials Hecht (1995) suggested setting this probability as the goal and used a computerized search method to determine optimal (by this criterion) fixed fractions for p  q = .02 and various c, n and specified success probabilities. This is a much easier problem than the similar sounding in Section 3.1. We have for the probability that X (T) at the end exceeds the goal: 1 = 1 P(X(T» ) aT+b)= ~ 1 v2rrT 1 .J2rrT where u= x/~ so x integral equals 00 exp/x 2 /2T}dx (/1'+1> 00 exp/u 2 / 2} du (/1'1 /2+1>1'  1/2 = aT +b gives u~ = aT +b and U = aTI /2 +bT  I/ 2. The 1  N(aTI /2 + bT  I/ 2) = N( _(aTI /2 + bT = 1 N(a + f3) 1/ 2)) = N(a  f3). (3 .3) For example (3 .1) f = .01l7 and P = .7947 . For example (3.2) P = .7433. Example (3.1) is for the Hecht optimal fraction and example (3 .2) is for the Kelly optimal fraction . Note the difference in P values. Our numerical results are consistent with Hecht's simulations in the instances we have checked. Browne (1996) has given an elegant continuous approximation solution to the problem: What is the strategy which maximizes the probability of reaching a fixed goal C on or before a specified time n and what is the corresponding probability of success? Note 800 E. O. Thorp 396 £.0. Thorp The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. that the optimal strategy will in general involve betting varying fractions, depending on the time remaining and the distance to the goal. As an extreme example, just to make the point, suppose n = I and C = 2. If Xo < 1 then no strategy works and the probability of success is O. But if I ~ Xo < 2 one should bet at least 2  Xo , thus any fraction f ? (2  Xo) / Xo, for a success probability of p. Another extreme example: n = 10, C = 2 10 = 1024, Xo = 1. Then the only strategy which can succeed is to bet f = 1 on every trial. The probability of success is pI 0 for this strategy and 0 for all others (if p < 1), including Kelly. 3.4. Continuous approximation of expected time to reach a goal According to Theorem I(v), the optimal growth strategy asymptotically minimizes the expected time to reach a goal. Here is what this means. Suppose for goal C that m(C) is the greatest lower bound over all strategies for the expected time to reach C. Suppose t *(C) is the expected time using the Kelly strategy. Then limc ,> 00 (t * (c) / m(c» = 1. The continuous approximation to the expected number of trials to reach the goal C > Xo = I is n(C, f) = (lnC) / g(f) where f is any fixed fraction strategy. Appendix C has the derivation. Now g(f) has a unique maximum at g(f*) so n(C, f) has a unique minimum at f = f *. Moreover, we can see how much longer it takes, on average, to reach C if one deviates from f*· 3.5. Comparing fixed fraction strategies: the probability that one strategy leads another after n trials Theorem I (iv) says that wealth using the Kelly strategy will tend, in the long run, to an infinitely large multiple of wealth using any "essentially different" strategy. It can be shown that any fixed f I f * is an "essentially different" strategy. This leads to the question of how fast the Kelly strategy gets ahead of another fixed fraction strategy, and more generally, how fast one fixed fraction strategy gets ahead of (or behind) another. If WI/ is the number of wins in n trials and n  WI/ is the number of losses, G(f) = (WI//n) In(l + f) + (1  WI//n) In(l  f) is the actual (random variable) growth coefficient. As we saw, its expectation is g(f) = E(G(f)) = p log(l + f) + q log(l  f) (3.4) and the variance of G(f) is VarG(f) = (pq) / n)lln((l + f) / (l f))}2 (3.5) and it follows that G(f), which has the form G(f) = a(L Tk) / n +b, is approximately normally distributed with mean g(f) and variance VarG(f). This enables us to give The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 801 Ch. 9: 397 The Kelly Criterion in Blackjack Sports Betting, and the Stock Market the distribution of X I1 and once again answer the question of Section 3.3. We illustrate this with an example. Example 3.4. = .51, q = .49, 1* = .02, s = standard deviation of G(f) The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. p N = 10,000 and gls 1 g s Pr(G(f) ( 0) 1.5 1.0 .01 .02 .03 .000150004 .000200013 .000149977 .0001 .0002 .0003 .067 .5 .159 .309 Continuing, we find the distribution of G(h)  GUt). We consider two cases. Case 1. The same game Here we assume both players are betting on the same trials, e.g., betting on the same coin tosses, or on the same series of hands at blackjack, or on the same games with the same odds at the same sports book. In the stock market, both players could invest in the same "security" at the same time, e.g., a noload S&P 500 index mutual fund. We find E(G(h)  G(It») = plog((1 + 12)/(1 + It») + q 10g((1 12)/(1  r and Var( G (h)  G (ft)) = (pq In) { log [ C 1\») ~ ~~) C ~ ~: ) J where G(h)  G(ft> is approximately normally distributed with this mean and variance. Case 2. Identically distributed independent games This corresponds to betting on two different series of tosses with the same coin. E(G(h)  G(fl» is as before. But now Var(G(h)  G(fj) = Var(G(h» + Var(G(fd) because G(h) and G(fj) are now independent. Thus Var( G(h)  G(It») = (pq In) {[IOgC ~ ~~) r + [IOgC ~ ~: rJ}· Let a = log ( 1 + 1\ ) , 1  It b = log (1 + h). 1 12 Then in Case 1, VI = (pq In)(a  b)2 and in Case 2, V2 = (pq In)(a 2 + b2) and since a , b > 0, VI . < V2 as expected. We can now compare the Kelly strategy with other The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. 802 E. O. Thorp 398 E.O. Thorp fixed fractions to determine the probability that Kelly leads after n trials. Note that this probability is always greater than 1/2 (within the accuracy limits of the continuous approximation, which is the approximation of the binomial distribution by the normal, with its well known and thoroughly studied properties) because g(f*)  g(f) > 0 where 1* = p  q and I =1= 1* is some alternative. This can fail to be true for small n, where the approximation is poor. As an extreme example to make the point, if n = 1, any I > f* beats Kelly with probability p > 1/ 2. If instead n = 2, I > f* wins with probability p2 and p2 > 1/ 2 if p > 1/.J2 ~ .707l. Also, I < f* wins with probability I  p2 and 1  p2 > 1/ 2 if p2 < 1/ 2, i.e., p < 1/ .J2 = .7071. So when n = 2, Kelly always loses more than half the time to some other I unless p = 1/.J2. We now have the formulas we need to explore many practical applications of the Kelly criterion. 4. The long run: when will the Kelly strategy "dominate"? The late John Leib wrote several articles for Blackjack Forum which were critical of the Kelly criterion. He was much bemused by "the long run". What is it and when, if ever, does it happen? We begin with an example. Example 4.1. p = .51 , n = 10,000, Vi and Si, i = 1, 2, are the variance and standard deviation, respectively, for Section 3.5 Cases I and 2, and R = V2/V[ = (a 2 + b 2)/(a  b)2 so S2 = s[/k Table I summarizes some results. We can also approximate v'"R with a power series estimate using JI? + 122/111  121· only the first term of a and of b: a ~ 2ft, b ~ 212 so v'"R == The approximate results , which agree extremely well, are 2.236, 3.606 and 1.581, respectively. The first two rows show how nearly symmetric the behavior is on each side of the optimal f* = .02. The column (g2  gl) / s[ shows us that f* = .02 only has a .5 standard deviation advantage over its neighbors I = .0 I and I = .03 after n = 10,000 Table I Comparing strategies  II .Ii g2  g l .1'1 C~2 .01 .03 .03 .02 .02 .00005001 .00005004 .00000003 .00010000 .00010004 .00020005 .50 .50 .00013346 .01 g l ) j.~ 1 ~ 2.236 3.604 1.581 The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 803 Ch. 9: 399 The Kelly Criterion in Blackjack Sports Betting, and the Stock Market The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. Table 2 The long run: Cli2  g I) /s after 11 trials II h 11 = 104 11=4 x 104 11=16 x 104 11 = 106 .01 .03 .03 .02 .02 .01 .5 .5 .000133 1.0 1.0 .000267 2.0 2.0 .000534 5.0 5.0 .001335 trials. Since this advantage is proportional to vIn, the column (g2  gl)ISj from Table 1 gives the results of Table 2. The factor JR in Table 1 shows how much more slowly h dominates fl in Case 2 versus Case 1. The ratio (g2  g I) 1S2 is JR times as large so the same level of dominance takes R times as long. When the real world comparisons of strategies for practical reasons often use Case 2 comparisons rather than the more appropriate Case I comparisons, the dominance of f* is further obscured. An example is players with different betting fractions at blackjack. Case 1 corresponds to both betting on the same sequence of hands. Case 2 corresponds to them playing at different tables (not the same table, because Case 2 assumes independence). (Because of the positive correlation between payoffs on hands played at the same table, this is intermediate between Cases 1 and 2.) It is important to understand that "the long run", i.e., the time it takes for f* to dominate a specified neighbor by a specified probability, can vary without limit. Each application requires a separate analysis. In cases such as Example 4.1, where dominance is "slow", one might argue that using f * is not important. As an argument against this, consider two cointossing games. In game 1 your edge is 1.0%. In game 2 your edge is 1.1 %. With one unit bets, after n trials the difference in expected gain is E2  E I = .00In with standard deviation s of about..JIn hence (E2  EI )Is ~ .001 vlnl v'2 which is 1 when n = 2 x 106 . So it takes two million trials to have an 84% chance of the game 2 results being better than the game 1 results. Does that mean it's unimportant to select the higher expectation game? S. Blackjack For a general discussion of blackjack, see Thorp (1962, 1966), Wong (1994) and Griffin (1979). The Kelly criterion was introduced for blackjack by Thorp (1962) . The analysis is more complicated than that of coin tossing because the payoffs are not simply one to one. In particular the variance is generally more than I and the Kelly fraction tends to be less than for coin tossing withthe same expectation. Moreover, the distribution of various payoffs depends on the player advantage. For instance the frequency of pair splitting, doubling down, and blackjacks all vary as the advantage changes. By binning the probability of payoff types according to ex ante expectation, and solving the Kelly equations on a computer, a strategy can be found which is as close to optimal as desired. The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. 804 E. 0. Thorp 400 E.O. Thorp There are some conceptual subtleties which are noteworthy. To illustrate them we'll simplify to the coin toss model. At each trial, we have with probability .5 a "favorable situation" with gain or loss X per unit bet such that P(X = I) = .51 , P(X = I) =.49 and with probability.5 an unfavorable situation with gain or loss Y per unit bet such that P(Y = I) = .49 and P(Y = 1) = .51. We know before we bet whether X or Y applies. Suppose the player must make small "waiting" bets on the unfavorable situations in order to be able to exploit the favorable situations. On these he will place "large" bets. We consider two cases. Case 1. Bet fa on unfavorable situations and find the optimal f* for favorable situations. We have g(f) = .5(.5110g(l + f) + .4910g(1  f)) + .5(.4910g(l + fa) + .51 log (I  fa»). (5.1) Since the second expression in (5 . 1) is constant, f maximizes g(f) if it maximizes the first expression, so f* = P  q = .02, as usual. It is easy to verify that when there is a spectrum of favorable situations the same recipe, f/ = Pi  qi for the ith situation, holds. Again, in actual blackjack f * would be adjusted down somewhat for the greater variance. With an additional constraint such as fi ( kfo, where k is typically some integral multiple of fo, representing the betting spread adopted by a prudent player, then the solution is just fi ( min(f/, kfo) . Curiously, a seemingly similar formulation of the betting problem leads to rather different results. Case 2. Bet f in favorable situations and af in unfavorable situations, 0 ( a ( I. Now the bet sizes in the two situations are linked and both the analysis and results are more complex. We have a Kelly growth rate of g(f) = .5(.5110g(1 + f) + .4910g(1  f)) + .5(.4910g(1 + af) + .5110g(l  af)) . (5.2) If we choose a = 0 (no bet in unfavorable situations) then the maximum value for g(f) is at f* = .02, the usual Kelly fraction. If we make "waiting bets", corresponding to some value of a > 0, this will shift the value of f* down, perhaps even to O. The expected gain divided by the expected bet is .02(1  a) / (l + a) , a ~ O. If a = 0 we get .02, as expected. If a = 1, we get 0, as expected: this is a fair game and the Kelly fraction is f* = O. As a increases from 0 to 1 the (optimal) Kelly fraction f* decreases from .02 to O. Thus the Kelly fraction for favorable situations is less in this case when bets on unfavorable situations reduce the overall advantage of the game. Arnold Snyder called to my attention the fact that Winston Yamashita had (also) made this point (March 18, 1997) on the "free" pages, miscellaneous section, of Stanford Wong's web site. The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 805 Ch. 9: 401 The Kelly Criterion in Blackjack Sports Betting, and the Stock Market The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. Table 3 f* versus a a 1'* a 1'* a 1'* 0 .1 .2 .3 .0200 .0178 .0154 .0 128 1/3 .4 .5 .6 .0120 .0103 .0080 .0059 .7 .8 .9 1.0 .0040 .0024 .0011 .0000 For this example, we find the new f* for a given value of a, 0 < a < 1, by solving g' (f) = O. A value of a = 1/3, for instance, corresponds to a bet of 1/3 unit on Y and I unit on X, a betting range of 3 to 1. The overall expectation is .0 I. Calculation shows /* = .01200l. Table 3 shows how /* varies with a. To understand why Cases I and 2 have different /*, look first at Equation (5.1). The part of g(f) corresponding to the unfavorable situations is fixed when fo is fixed. Only the part of g(f) corresponding to the favorable situations is affected by varying f. Thus we maximize g(f) by maximizing it over just the favorable situations. Whatever the result, it is then reduced by a fixed quantity, the part of g containing fo. On the other hand, in Equation (5.2) both parts of g(f) are affected when f varies, because the fraction af used for unfavorable situations bears the constant ratio a to the fraction f used in favorable situations. Now the first term, for the favorable situations, has a maximum at f = .02, and is approximately "flat" nearby. But the second term, for the unfavorable situations, is negative and decreasing moderately rapidly at f = .02. Therefore, it we reduce f somewhat, this term increases somewhat, while the first term decreases only very slightly. There is a net gain so we find f* < .02. The greater a is, the more important is the effect of this term so the more we have to reduce f to get f*, as Table 3 clearly shows. When there is a spectrum of favorable situations the solution is more complex and can be found through standard multivariable optimization techniques. The more complex Case 2 corresponds to what the serious blackjack player is likely to need to do in practice. He will have to limit his current maximum bet to some multiple of his current minimum bet. As his bankroll increases or decreases, the corresponding bet sizes will increase or decrease proportionately. 6. Sports betting In 1993 an outstanding young computer science Ph.D. told me about a successful sports betting system that he had developed. Upon review I was convinced. I made suggestions for minor simplifications and improvements. Then we agreed on a field test. We found a person who was extremely likely to always be regarded by the other sports bettors as a novice. I put up a test bankroll of $50,000 and we used the Kelly.system to estimate our bet size. 806 E. O. Thorp 402 £.0. Thorp +80,000.00 +70,000.00 1 I Total Profit  Expected Profit I +60.000.00 +50,000.00 +40, 000.00 +30.000.00 The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. +20,000.00 t. +10,000.00 . +0 .00 3 1 5 7 II ~ 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43  10,000.00 Fig. 3. Betting log Type 2 sports. +70. 000.00 +60,000.00 H I Total Profit       Expected Profit I +50.000.00 .... / +40.000.00 1 .... ~ +30, 000.00 +20,000.00 . ~  . ..  .  +10,000.00 .. ..    ~ _I, +0.00 1 5 I 9 13 17 2 1 25 29 33 37 41 45 49 53 57 61 65 69 73 77 8 1 85 89 93 97 101 1 0 .000.00 Fig. 4. Betting log Type I sports. We bet on 101 days in the first four and a half months of 1994. The system works for various sports. The results appear in Figures 3 and 4. After 101 days of bets, our $50,000 bankroll had a profit of $ 123,00Q, about $68,000 from Type I sports and about $55,000 from Type 2 sports. The expected returns are shown as about $62,000 for Type 1 and about $27,000 for Type 2. One might assign the additional $34,000 actuall y won to luck. But this is likely to be at most partly true because our expectation estimates from the model were deliberately chosen to be conservative. The reason is that usi ng too large an f* and overbetting is much more severely penalized than using too small an f * and underbetting. . The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 807 efl. 9: 403 The Kelly Criterion in Blackjack Sports Betting, and th e Stock Market Though $123,000 is a modest sum for some, and insignificant by Wall Street standards, the system performed as predicted and passed its test. We were never more than a few thousand behind. The farthest we had to invade our bankroll to place bets was about $10,000. Our typical expectation was about 6% so our total bets ("action") were about $2,000,000 or about $20,000 per day. We typically placed from five to fifteen bets a day and bets ranged from a few hundred dollars to several thousand each, increasing as our bankroll grew. Though we had a net win, the net results by casino varied by chance from a substantial loss to a large win. Particularly hard hit was the "sawdust joint" Little Caesar's. It "died" towards the end of our test and I suspect that sports book losses to us may have expedited its departure. One feature of sports betting which is of interest to Kelly users is the prospect of betting on several games at once. This also arises in blackjack when (a) a player bets on multiple hands or (b) two or more players share a common bankroll. The standard techniques readily solve such problems. We illustrate with: Example 6.1. Suppose we bet simultaneously on two independent favorable coins with betting fractions It and 12 and with success probabilities PI and P2, respectively. Then the expected growth rate is given by gUI, h) + II + h) + Plq2 ln (1 + It  h) + QlP2 ln (l  It + h) + QlQ2 ln (l  II  h)· = PIP2 In (1 it To find the optimal and ilg/a12 = O. The result is I I + f 2  PI P2  q I q2  PIP2 It  12 H we solve the simultaneous equations ag/alt + qlq2 = PI q2  ql P2 Plq2 qlP2 it = (c + d)/2, + = = 0 and c, == d, H = (c  (6.1) d)/2. These equations pass the symmetry check: interchanging 1 and 2 throughout maps the equation set into itself. An alternate form is instructive. Let mi = Pi  qi, i = 1, 2 so Pi = (1 + mj) /2 and qi = (1  mj)/2 . Substituting in (6.1) and simplifying leads to: ml +m2 c=l+mlm2 ' * II = ml(lm~) (6.2) I m 2 m 2 ' l 2 which shows clearly the factors by which the It are each reduced from m7 Since the mj are typically a few percent, the reduction factors are typically very close to 1. 808 E. O. Thorp 404 E.O. Thorp In the special case PI = P2 = p, d = 0 and 1* = = f; = e/2 = (p  q)/ (2(p2 + q2». Letting m = p  q this may be written 1* = m/(l + m 2) as the optimal fraction to bet on each coin simultaneously, compared to f* = m to bet on each coin sequentially. Our simultaneous sports bets were generally on different games and typically not numerous so they were approximately independent and the appropriate fractions were only moderately less than the corresponding single bet fractions. Question: Is this always true for independent simultaneous bets? Simultaneous bets on blackjack hands at different tables are independent but at the same table they have a pairwise correlation that has been estimated at .5 (Griffin, 1979, p. 142). This should substantially reduce the Kelly fraction per hand. The blackjack literature discusses approximations to these problems. On the other hand, correlations between the returns on securities can range from nearly 1 to nearly 1. An extreme correlation often can be exploited to great advantage through the techniques of "hedging". The risk averse investor may be able to acquire combinations of securities where the expectations add and the risks tend to cancel. The optimal betting fraction may be very large. The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. it The next example is a simple illustration of the important effect of covariance on the optimal betting fraction. Example 6.2. We have two' favorable coins as in the previous example but now their outcomes need not be independent. For simplicity assume the special case where the two bets have the same payoff distributions, but with ajoint distribution as in Table 4. Now c+m +b = (l +m)/2 so b = (lm)/2 c and therefore 0 ::;; e ::;; (lm)/2. Calculation shows Var(Xi) = 1  m 2, Cor(XI, X2) = 4c  (1  m)2 and Cor(XI, X2) = [4c  (1  m)2]/(l  m 2). The symmetry of the distribution shows that g(fl, h) will have its maximum at fl :;= h = f so we simply need to maximize g(f) = (c + m) In(1 + 2f) + e In(l  2f). The result is 1* = m/(2(2c + m». We see that for m fixed, as c decreases from (l  m)/2 and cor(X I , X2) = 1, to 0 and cor(X I, X2) = (1  m)/(1 + m), 1* for each bet increases from m/2 to 1/2, as in Table 5. Table 5 Table 4 Joint distribution of two "identical" favorable coins with correlated outcomes I* increases as Cor( X I . X 2) decreases Cor(XI. X2) XI: X2: I I C+1I1 h c I* C (I  m) / 2 I b 0 (I  111) / (1 (I  + 111) 0 /11 2)/ 4 111 / 2 111 / (1 +111 2 ) 1/ 2 The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 809 ('II. 9: 405 The Kelly Criterion in Blackjack Sports Betting, and the Stock Market It is important to note that for an exact solution or an arbitrarily accurate numerical approximation to the simultaneous bet problem, covariance or correlation information is not enough. We need to use the entire joint distribution to construct the g function . We stopped sports betting after our successful test for reasons including: (I) It required a person on site in Nevada. (2) Large amounts of cash and winning tickets had to be transported between casinos. We believed this was very risky. To the sorrow of others, subsequent events confirmed this. (3) It was not economically competitive with our other operations. If it becomes possible to place bets telephonically from out of state and to transfer the corresponding funds electronically, we may be back. 7. Wall street: the biggest game To illustrate both the Kelly criterion and the size of the securities markets, we return to the study of the effects of correlation as in Example 6.2. Consider the more symmetric and esthetically pleasing pair of bets VI and V2 , with joint distribution given in Table 6. Clearly 0 :::;; a :::;; 1/ 2 and Cor( VI , V2) = Cor( V I, Vz:) = 4a  I increases from 1 to 1 as a increases from 0 to 1/ 2. Finding a general solution for (ft, 1;) appears algebraically complicated (but specific solutions are easy to find numerically), which is why we chose Example 6.2 instead. Even with reduction to the special case m 1 = m2 = m and the use of symmetry to reduce the problem to finding f* = ft = f;, a general solution is still much less simple. But consider the instance when a = 0 so Cor(VI , V2) = \. Then g(f) = In(l + 2m!) which increases without limit as f increases. This pair of bets is a "sure thing" and one should bet as much as possible. This is a simplified version of the classic arbitrage of securities markets: find a pair of securities which are identical or "equivalent" and trade at disparate prices. Buy the relatively underpric~d security and sell short the relatively overpriced security, achieving a correlation of I and "locking in" a riskless profit. An example occurred in 1983. My investment partnership bought $ 330 million worth of "old" AT&T and sold short $332.5 million worth of whenissued "new" AT&T plus the new "seven sisters" regional telephone companies . Much of this was done in a single trade as part of what was then the largest dollar value block trade ever done on the New York Stock Exchange (December 1, 1983). In applying the Kelly criterion to the securities markets, we meet new analytic problems. A bet on a security typically has many outcomes rather than just a few, as in Table 6 Joint di stributi on of VI and V 2 11/ 2  1 11/1 + 1 (/ 1/ 2  ({ 11/1  I 1/ 2  ({ ({ The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. 810 E. O. Thorp 406 E.O. Thorp most gambling situations. This leads to the use of continuous instead of discrete probability distributions. We are led to find f to maximize g(f) = E In(l + f X) = f In(l + fx) dP(x) where P(x) is a probability measure describing the outcomes. Frequently the problem is to find an optimum portfolio from among n securities, where n may be a "large" number. In this case x and fare ndimension vectors and f x is their scalar product. We also have constraints. We always need I + fx > 0 so In( . ) is defined, and L fi = I (or some c > 0) to normalize to a unit (or to a c > 0) investment. The maximization problem is generally solvable because g(f) is concave. There may be other constraints as well for some or all i such as fi ~ 0 (no short selling), or fi ~ Mi or fi ~ mi (limits amount invested in ith security), or L Ifi! ~ M (limits total leverage to meet margin regulations or capital requirements). Note that in some instances there is not enough of a good bet or investment to allow betting the full f* , so one is forced to underbet, reducing somewhat both the overall growth rate and the risk. This is more a problem in the gaming world than in the much larger securities markets. More on these problems and techniques may be found in the literature. 7. 1. Continuous approximation There is one technique which leads rapidly to interesting results . Let X be a random variable with P(X = m + s) = P(X = m  5) = .5. Then E(X) = m, Var(X) = 52. With initial capital Vo, betting fraction f, and return per unit of X, the result is V(f) = Vo(1 + (I  f)r + f X) = Vo(1 + r + f(X  r»), where r is the rate of return on the remaining capital, invested in, e.g., Treasury bills. Then g(f) = E(G(f») = E(ln(V(f)1 Vo)) = E In(1 + r + f(X = .5In(1 + r + f(m  r + s») + .5In(1 + r + f(m   r») r  s»). Now subdivide the time interval into n equal independent steps, keeping the same drift and the same total variance. Thus m, 52 and r are replaced by min, 52 I nand r I n, respectively. We have n independent Xi, i = I , .. . , n, with P(Xi =mln + sn I / 2) = P(Xi = min  V,,(f)IVo = n(1 + (I sn I / 2 ) = .5. Then II  f)r + fXi). i=1 Taking E(log(·» of both sides gives g(f). Expanding the result in a power series leads to (7.1 ) where 0(n  I / 2 ) has the property n 1/ 20(n  I/ 2) is bounded as n ~ 00. Letting n ~ 00 in (7. 1) we have goo (f) == r + f(m  r)  s2 f2 /2 . (7.2) The Kelly Criterion in Blackjack Sports Betting, and the Stock Market The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. ('iI. 9: The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 811 407 The limit V == V00 (f) of Vii (f) as n + 00 corresponds to a log normal diffusion process, which is a wellknown model for securities prices. The "security" here has instantaneous drift rate m, variance rate s2, and the riskless investment of "cash" earns at an instantaneous rate r. Then goo (f) in (7.2) is the (instantaneous) growth rate of capital with investment or betting fraction f. There is nothing special about our choice of the random variable X. Any bounded random variable with mean E(X) = m and variance Var(X) = s2 will lead to the same result. Note that f no longer needs to he less than or equal to 1. The usual problems, with log(·) being undefined for negative arguments, have disappeared. Also, f < 0 causes no problems. This simply corresponds to selling the security short. If m < r this could be advantageous. Note further that the investor who follows the policy f must now adjust his investment "instantaneously". In practice this means adjusting in tiny increments whenever there is a small change in V. This idealization appears in option theory. It is well known and does not prevent the practical application of the theory (Black and Scholes, 1973). Our previous growth functions for finite sized betting steps were approximately parabolic in a neighborhood of f* and often in a range up to 0 ~ f ~ 2f*, where also often 2f* == t. Now with the limiting case (7.2), goo (f) is exactly parabolic and very easy to study. Lognormality of V (f) / Vo means 10g(V Vo) is N (M, 52) distributed, with mean M = goo(nt and variance 52 = Var(Goo(f)t for any time t. From this we can determine, for instance, the expected capital growth and the time tk required for V (f) to be at least k standard deviations above Vo. First, we can show by our previous methods that Var(Goo(f» = s2 f2, hence Sdev(Goo(f» = sf. Solving tkgoo = kt} /2 Sdev(Goo(f» gives tkg~ hence the expected capital growth tkgoo, from which we find tk. The results are summarized in Equations (7.3). (n / f* = (m  r)/s2, goo(f*) = (m  r)2/2s2 Var(Goo(f») = tkgoo(f) goo(f) = r s2 f2, + f(m  r)  s2 f2 /2, + r, Sdev(Goo(f») = sf, = k 2s 2 f2 / goo, tk = k 2s 2 12 / g~. (7.3) Examination of the expressions for tk goo (f) and tk show that each one increases as f increases, for 0 ~ 1 < 1+ where f + is the positive root of s2 f2 /2  (m  r) 1  r = 0 and 1+ > 2f*. Comment: The capital asset pricing model (CAPM) says that the market portfolio lies on the Markowitz efficient frontier E in the (s, m) plane at a (generally) unique point P = (so, mo) such that the line determined by P and (s = 0, m = r) is tangent to E (at P). The slope of this line is the Sharpe ratio 5 = (mo  ro)/so and from (7.3) Koo(f*) = 5 2 /2 + r so the maximum growth rate goo(f*) depends, for fixed r, only on the Sharpe ratio. (See Quaife (1995).) Again from (7.3), f* = 1 when m = r + s2 in which case the Kelly investor will select the market portfolio without borrowing or lending. If m > r + s2 the Kelly investor will use leverage and if m < r + s2 he will 812 E. 0. Thorp 408 E.O. Tho!]) The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. invest partly in Tbills and partly in the market portfolio. Thus the Kelly investor will dynamically reallocate as f* changes over time because of fluctuations in the forecast m, rand s2, as well as in the prices of the portfolio securities. From (7.3), goo(1) = m  s2/2 so the portfolios in the (s, m) plane satisfying m  s2/2 = C, where C is a constant, all have the same growth rate. In the continuous approximation, the Kelly investor appears to have the utility function U (s, m) = m  s2/2. Thus, for any (closed, bounded) set of portfolios, the best portfolios are exactly those in the subset that maximizes the one parameter family m  s2/2 = C. See Kritzman (1998), for an elementary introduction to related ideas. Example 7.1. The long run revisited. For this example let r = O. Then the basic equations (7 .3) simplify to 1* = m/s2 , goo(f*) = m 2 /2s2, Var(G oo (f») = s2 f2, r = 0: goo(f) = mf  s2 f2/2, (7.4) Sdev(Goo(f») = sf. How long will it take for V (f*) ~ Vo with a specified probability? How about V (f* /2)? To find the time t needed for V (f) ~ Vo at the k standard deviations level of significance (k = I, P = 84%; k = 2, P = 98%, etc.) we solve for t == tk: (7.5) We get more insight by normalizing all f with f* · Setting f find when r = 0 = c f* throughout, we == m/s2 , f = em/s2 , goo(c1*) = m 2 (c  c2 /2)/s 2 , Sdev(G oo (c1*») = cm/s, r = 0: f* tg oo (c1*) = k 2e/(1  e/2), t(k ,ef*) =k 2s 2 /{m 2 (1_e/2)2). (7 .6) Equations (7.6) contain a remarkable result: V (f) ~ Vo at the k standard deviation level of significance occurs when expected capital growth tgoo = k 2 e/(1  e/2) and this result is independent of m and s. For f = f* (e = 1 in (7.6», this happens for k = I at tgoo = 2 corresponding to V = Voe 2 and at k = 2 for tgoo = 8 corresponding to V = Voe 8 . Now e 8 === 2981 and at a 10% annual (instantaneous) growth rate, it takes 80 years to have a probability of 98% for V ~ Yo. At a 20% annual instantaneous rate it takes 40 years. However, for f = f* /2, the number for k = I and 2 are tgoo = 2/3 and 8/3, respectively, just 1/3 as large. So the waiting times for Prob(V ~ Yo) to exceed 84% and 98% become 6.7 years and 26.7 years, respectively, and the expected growth rate is reduced to 3/4 of that for f*. The Kelly Criterion in Blackjack Sports Betting, and the Stock Market ('iI. 9: 813 409 The Kelly Criterion in Blackjack Sports Betting, and the Stock Marker Comment: Fractional Kelly versus Kelly when r = 0 From Equations (7,6) we see that goo(cf*)/goo(f*) = c(2  c), 0 The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. ~ c < 00, showing how the growth rate relative to the maximum varies with c, The relative risk Sdev(Goo(cf*»/Sdev(Goo(f*» = c and the relative time to achieve the same expected total growth is 1/c(2  c), 0 < c < 2, Thus the relative "spread" for the same expected total growth is 1/(2  c), 0 < c < 2. Thus, even by choosing c very small, the spread around a given expected growth cannot be reduced by 1/2, The cOlTesponding results are not quite as simple when r > O. 7.2. The (almost) real world Assume that prices change "continuously" (no "jumps"), that portfolios may be revised "continuously", and that there are no transactions costs (market impact, commissions, "overhead"), or taxes (Federal, State, city, exchange, etc.). Then our previous model applies. Example 7.2. The S&P 500 Index. Using historical data we make the rough estimates = .11, s = .15, r = .06. The equations we need for r =1= 0 are the generalizations of (7.6) to r =1= 0 and f = cf*, which follow from (7.3): 111 cf* = c(m  r)/s2, goo(cf*) = (m  r)2(c  c 2/2))/s2 + r, Sdev(Goo(cf*») = c(m  r)/s, tgoo(cf*) = k 2c 2/(c  c 2/2 + rs 2/(m  r)2) , t(k, cf*) = k 2c 2 ( (m  r)2 /s2) / (({m  r)2/s 2)(c  c 2/2) + r( (7.7) If we define iii = m  r, Goo = Goo  r, goo = goo  r, then substitution into Equations (7.7) give Equations (7.6), showing the relation between the two sets. It also shows that examples and conclusions about P (Vn > Vo) in the r = 0 case are equivalent to those about P(ln(V(t)/Vo) > rt) in the r =1= 0 case. Thus we can compare various strategies versus an investment compounding at a constant riskless rate r such as zero coupon U.S. Treasury bonds. ' From Equations (7.7) and c = 1, we find f* = 2.22, goo(f*) = .115, Sdev( Goo(f*») = .33, t = 8.32k 2 years. Thus, with f* = 2.22, after 8.32 years the probability is 84% that VII > Vo and the expected value of loge VII / Vo) = .96 so the median value of VII / Vo will be about £, . ,96 = 2.61. With the usual unlevered f = 1, and c = .45, we find using (7.3) goo(1) = m  s2/2 = .09875, Sdev( Goo(l») = .15, t(k, .45f*) = 2.31k 2 years. 814 E. 0. Thorp 410 E.O. Thorp Writing tgoo = h(c) in (7.7) as we see that the measure of riskiness, h(c) , increases as c increases, at least up to the point c = 2, corresponding to 2f* (and actually beyond, up to 1 + Writing t(k, c1*) = t(c) as The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. t(c) = k 2 «m  r)2/s2)/(m  r)2/s2)(l  c/2) 1 + (1/:~~. )1 ). + r/c 2 shows that t(c) also increases as c increases, at least up to the point c = 2. Thus for smaller (more conservative) f = cf*, c ( 2, specified levels of P(VI/ > Vo) are reached earlier. For c < 1, this comes with a reduction in growth rate, which reduction is relatively small for f near 1* . Note: During the period 19751997 the short term Tbill total return for the year, a proxy for r if the investor lends (i.e., f < 1), varied from a low of 2.90% (1993) to a high of 14.71 % (1981). For details, see Ibbotson Associates, 1998 (or any later) Yearbook. A large well connected investor might be able to borrow at broker's call plus about 1%, which might be approximated by Tbills plus I %. This might be a reasonable estimate for the investor who borrows (f > 1). For others the rates are likely to be higher. For instance the prime rate from 19751997 varied from a low of 6% (1993) to a high of 19% (1981), according to Associates First Capital Corporation (1998). As r fluqtuates , we expect m to tend to fluctuate inversely (high interest rates tend to depress stock prices for well known reasons). Accordingly, f* and goo will also fluctuate so the 'long term S&P index fund investor needs a procedure for periodically reestimating and revising f* and his desired level of leverage or cash. To illustrate the impact of r" > r, where r" is the investor's borrowing rate, suppose r" in example (7.2) is r + 2% or .08, a choice based on the above cited historical values for r, which is intermediate between "good" rt, ~ r + I %, and "poor" r" ~ the pri~e rate ~ r + 3%. We replace r by rt, in Equations (7.7) and, if f* > I, f* = 1.33, goo(f*) = .100, Sdev(Goo(f*» = .20, tgoo(f*) = .4k2 , t = 4k2 years. Note how greatly f* is reduced. Comment: Taxes Suppose for simplicity that aU gains are subject to a constant continuous tax rate T and that all losses are subject to a constant continuous tax refund at the same rate T. Think of the taxing entities, collectively, as a partner that shares a fraction T of all gains and losses. Then Equations (7 .7) become: c1* = c(m  r) /s2 0  T), goo(c.t*) = (m  r)2(c  c 2 /2))/s2 Sdev(Goo(c.t*») = c(m  r)/s, + r( I  T), The Kelly Criterion in Blackjack Sports Betting, and the Stock Market ( 'II. <J: The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 815 411 tgoo (cf*) = k 2 c2 / {c  c 2/ 2 + r(1  T)s2 / (m  r2)) , t(k, cf*) = k 2c 2((m  r)2 / s2)j{((m  r)2 / s2)(c  c2/2) +r(1 T)f The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. (7.7T) It is interesting to see that cf* increases by the factor I/O  T) . For a high income California resident, the combined state and federal marginal tax rate is 45 % so this ractor is 1/. 55 = 1.82. The amplification of cf* leads to the same growth rate as before exeept for a reduction by rT . The Sdev is unchanged and t (k, cf*) is increased slightly. However, as a practical matter, the much higher leverage needed with a high tax rate is typically not allowed under the margin regulation or is not advisable because the inability to continuously adjust in the real world creates dangers that increase rapidly with the degree of leverage. 7.3. The case for "fractional Kelly " Figure 5 shows three g curves for the true m : mr = .5m e , LOme and 1.5m e , where is the estimated value of m. The vertical lines and the slanting arrows illustrate the n:duction in g for the three choices of: f = .5 f e*' f e* and 1.5 f e* ' For example with f = .5fe* or "half Kelly", we have no loss and achieve the maximum g = .25 , in case mr = .5m e . But if mr = me then g = .75, a loss of .25 and if mr = 1.5m e then g = 1.25, a loss of 1.0, where all g are in units of m;/2s2. This is indicated both by I.aSSl and LOSS2 on the vertical line above f / j~* = .5, and by the two corresponding arrows which lead upward, and in this case to the right, from this line. A disaster occurs when mr = .5m e but we choose f = l.5 f e* ' This combines overbetting f "* by 50% with the overestimate of me = 2mr. Then g = .75 and we will be ruined. It is still had to choose f = f e* when mr = .5m e for then g = 0 and we suffer increasingly wild oscillations, both up and down, around our initial capitf\l. During a large downward oscillation experience shows that bettors will generally either quit or be eliminated by a minimum bet size requirement. Some lessons here are: ( I) To the extent me is an uncertain estimate of mr, it is wise to assume m{ < me and to choose f < f e* by enough to prevent g ~ O. Estimates of me in the stock market have many uncertainties and, in cases of forecast excess return, are more likely to be too high than too low. The tendency is to regress towards the mean . Securities prices follow a "nonstationary process" where m and s vary somewhat unpredictably over time. The economic situation can change for companies, industries, or the economy as a whole. Systems that worked may be partly or entirely based on data mining so m { may be substantially less than me. Changes in the "rules" such as commissions, tax laws, margin regulations, insider trading laws, etc., can also affect mr . Systems that do work attract capital, which tends to push exceptional mt down towards average values. The drift down means me > mr is likely. III, . 816 E. 0. Thorp 412 £.0. Thorp g(f) unit ~ m'/2s' PENALTY FOR EACH m, IFYOU CHOOSE THIS r r  o Sf: r   Tf :  1 Sf: 2.0 The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. 1.5 1.0 0 .5 0.0 ++':__\! !}'Ht..+ If: 2:0 LOSS, 0.5  1.0     .      ..   .  . Fig_5_ Penalties for choosing f = j;, i I* = ji. Sports betting has much the same caveats as the securities markets, with its own differences in detail. Rules changes, for instance, might include: adding expansion teams; the three point rule in basketball ; playing overtime sessions to break a tie; changing types of bats, balls, gloves, racquets or surfaces. Blackjack differs from the securities and sports betting markets in that the probabilities of outcomes can in principle generally be either calculated or simulated to any desired degree of accuracy. But even here ml is likely to be at least somewhat less than me. Consider player fatigue and errors, calculational errors and mistakes in applying either blackjack theory or Kelly theory (e.g. , calculating f* correctly, for which some of the issues have been discussed above), effects of a fixed shuffle point, nonrandom shuffling, preferential shuffling, cheating, etc. (2) Subject to (I), choosing f in the range .5.ft; ~ f < f {; offers protection against g ~ 0 with a reduction of g that is likely to be no more 'than 25%. The Kelly Criterion in Blackjack Sports Betting, and the Stock Market The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. ( '/1 , <J: 817 The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 4t3 Example 7.3. The great compounder, In 1964 a young hedge fund manager acquired a \uhstantial interest in a small New England textile company called Berkshire Hathaway. The stock traded then at 20. In 1998 it traded at 70,000, a multiple of 3500, and an annualized compound growth rate of about 27%, or an instantaneous rate of 24%. The once young hedge fund manager Warren Buffett is now acknowledged as the greatest investor or our time, and the world's second richest man. You may read about Buffett in Buffett and Clark (1997), Hagstrom (1994, 2004), Kilpatrick (1994) and Lowenstein (1995). If, as 1 was, you were fortunate enough to meet Buffett and identify the Berkshire opportunilY, what strategy does our method suggest? Assume (the somewhat smaller drift rate) III = .20, s = .15, r = .06. Note: Plausible arguments for a smaller future drift rate include regression towards the mean, the increasing size of Berkshire, and risk from the aging of management. A counterargument is that Berkshire's compounding rate has heen as high in its later years as in its earlier years. However, the S&P 500 Index has performed much better in recent years so the spread between the growth rates of the Index and of Berkshire has been somewhat less. So, if we expect the Index growth rate to revert towards the historical mean, then we expect Berkshire to do so even more. From Equations (7.3) or (7.7), f* = 6.22, goo(f*) = .495, t = Sdev(Goo(f*») = .93, 3.54k 2 years. Compare this to the unlevered portfolio, where lind: f = 1, goo(f) = .189, tk f = 1 and c = 1/6.22 === .1607. We Sdev( Goo (f) ) = .15, = .63k 2 years. Leverage to the level 6.22 would be inadvisable here in the real world because securities prices may change suddenly and discontinuously. In the crash of October, 1987, the S&P 500 index dropped 23% in a single day. If this happened at leverage of 2.0, the new leverage would suddenly be 77 /27 = 2.85 before readjustment by selling part of the portfolio. In the case of Berkshire, which is a large welldiversified portfolio, suppose we chose the conservative f = 2.0. Note that this is the maximum initial leverage allowed "customers" under current regulations. Then goo(2) = .295. The values in 30 years for median VXl/ Vo are approximately: f = I, Voo / Vo = 288; f = 2, V00/ Vo = 6, 974; f = 6.22, V00/ Vo = 2.86 X 106 . So the differences in results of leveraging are enormous in a generation. (Note: Art Quaife reports s = .24 for 19801997. The reader is invited to explore the example with this change.) The results of Section 3 apply directly to this continuous approximation model of a (possibly) leveraged securities portfolio. The reason is that both involve the same "dynamics", namely log Gil (f) is approximated as (scaled) Brownian motion with drift. So we can answer the same questions here for our portfolio that were answered in Section 3 for casino betting. For instance (3 .2) becomes Prob(V (t)/ Vo ~ x for some t) = x '" (2goo/ Var(G oo ») (7.8) 818 E. O. Thorp 414 E.O. Thorp where 1\ means exponentiation and 0 < x < 1. Using 0.4), for r 2g oo /Var(G oo ) = 1 so this simplifies to ProbO = 0 and f = x. = f* , 0 ·9) The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. Compare with Example 3.3 For 0 < r < m and f = f* the exponent of x in (7 .9) becomes I + 2rs 2/ (m  r)2 and has a positive first derivative so, as r increases, PO decreases since 0 < x < 1, tending to 0 as r tends to m, which is what we expect. 7.4. A remarkable formula In earlier versions of this chapter the exponent in Equations (3.2), 0 .8) and (7.9) were off by a factor of 2, which I had inadvertently dropped during my derivation. Subsequently Don Schlesinger posted (without details) two more general continuous approximation formulas for the r = 0 case on the Internet at www.bjmath.com dated June 19, 1997. If Vo is the initial investment and y > I > x > 0 then for /* the probability that V (t) reaches y Vo before x Vo is Prob(V (t, f*) reaches y Vo before x Vo) = (I  x) / (I  (x / y») (7.10) and more generally, for f = cf*, 0 < c < 2, Prob(V(t , cf*) reaches y VO before xVo) = [I  / \ (2 / c ~ 1)]/[1  (x/y(' (2/ c  I)] (7 . 11 ) where 1\ means exponentiation . Clearly (7.10) follows from 0 . 11) by choosing c = I. The r = 0 case of our Equation (7.8) follows from 0 . 11) and the r = 0 case of our Equation (7.9) follows from (7.10). We can derive a generalization of (7.1 I) by using the classical gambler's ruin formula (Cox and Miller, 1965, p. 31, Equation (2.0» and passing to the limit as step size tends to zero (Cox and Miller, 1965, pp. 205206), where we think of 10g(V(t, f)/ Vo) as following a diffusion process with mean goo and variance v(G oo ), initial value 0, and absorbing barriers at log y and log x . The result is Prob(V(t, cf*) reaches yVO before xVo) = [I  x Aa]/[1  (x/y) Aa] (7. 12) where a = 2goo / V (G (0) = 2M / V where M and V are the drift and variance, respectively, of the diffusion process per unit time. Alternatively, (7.12) is a simple restatement of the known solution for the Wiener process with two absorbing barriers (Cox and Miller, 1965, Example 5.5). As Schlesinger notes, choosing x = 1/2 and y = 2 in (7.10) gives probe V (t , /*) doubles before halving) = 2/3. Now consider a gambler or investor who focuses only on values VII = 211 Vo. n = o. ± I. ±2, . . . , multiples of his initial capital. In log space, loge VII / Vo) = n log 2 so we have a random walk on the integer multiples of log 2, where The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 819 The Kelly Criterion in Blackjack Sports Betting, and the Stock Market ( "II (): 415 tlie probability of an increase is p = 2/3 and of a decrease, q = 1/3. This gives us a lItvcnient compact visualization of the Kelly strategy's level of risk. If instead we choose c = 1/2 ("half Kelly"), Equation (7.11) gives probe V (t, !* /2) douhles before halving) = 8/9 yet the growth rate goo(f* /2) = .75g oo (f*) so "half Kelly" has 3/4 the growth rate but much less chance of a big loss. A second useful visualization of comparative risk comes from Equation (7 .8) which l( (7.13) The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. Prob(V(t,c!*)/Vo ::;;x forsomet) = xl\(2/c l). I'or c = 1 we had Prob(·) = x and for c = 1/2 we get Prob(·) = x 3 . Thus "half Kelly" has a much lessened likelihood of severe capital loss. The chance of ever losing lialf the starting capital is 1/2 for f = !* but only 1/8 for f = !* /2. My gambling alld investment experience, as well as reports from numerous blackjack players and Il'ams, suggests that most people strongly prefer the increased safety and psychological l'Olllfort of "half Kelly" (or some nearby value), in exchange for giving up 1/4 of their gn lwth rate. X. A case study III the summer of 1997 the XYZ Corporation (pseudonym) received a substantial alllount of cash. This prompted a review of its portfolio, which is shown in Table 7 iII the column 8/17/97 . The portfolio was 54% in Biotime, ticker BTIM, a NASDAQ hiotechnology company. This was due to existing and historical relationships between people in XYZ Corp. and in BTIM. XYZ's officers and directors were very knowledgeTable 7 Statistics for logs of monthly wealth relatives , 3/31/92 through 6/30/97 Berkshire BioTime SP500 Tbills Monthly Mean Standard deviation .0264 .0582 .0186 .2237 .0146 .0268 .0035 .0008 Allllual Mean Standard deviation .3167 .2016 .2227 .7748 .l753 .0929 .0426 .0028 Monthly Covariance .0034 .0021 .0500 .0005 .0001 .0007 1.2E06 3.2E05 5.7E 06 6.7E07 Monthly Correlation 1.0000  .1581 1.0000 .2954 .0237 1.0000 .0257 .1773 .2610 1.0000 820 E. O. Thorp 416 E.O. Thorp able about BTIM and felt they were especially qualified to evaluate it as an investment. They wished to retain a substantial position in BTIM. The portfolio held Berkshire Hathaway, ticker BRK, having first purchased it in 1991. The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. 8.1. The constraints Dr. Quaife determined the Kelly optimal portfolio for XYZ Corp. subject to certain constraints. The list of allowable securities was limited to BTIM, BRK, the Vanguard 500 (S&P 500) Index Fund, and Tbills. Being short Tbills was used as a proxy for margin debt. The XYZ broker actually charges about 2% more, which has been ignored in this analysis. The simple CAPM (capital asset pricing model) suggests that the investor only need consider the market portfolio (for which the S&P 500 is being substituted here, with well known caveats) and borrowing or lending. Both Quaife and the author were convinced that BRK was and is a superior alternative and their knowledge about and long experience with BRK supported this. XYZ Corp. was subject to margin requirements of 50% initially and 30% maintenance, meaning for a portfolio of securities purchased that initial margin debt (money lent by the broker) was limited to 50% of the value of the securities, and that whenever the value of the account net of margin debt was less than 30% of the value of the securities owned, then securities would have to be sold until the 30% figure was restored. In addition XYZ Corp. wished to continue with a "significant" part of its portfolio in BTIM. 8.2. The analysis and results Using monthly data from 3/31/92 through 6/30/97, a total of 63 months, Quaife finds the means, covariances, etc. given in Table 7. Note from Table 7 that BRK has a higher mean and a lower standard deviation than BTIM, hence we expect it to be favored by the analysis . But note also the negative correlation with BTIM, which suggests that adding some BTIM to BRK may prove advantageous. Using the statistics from Table 7, Quaife finds the following optimal portfolios, under various assumptions about borrowing. As expected, BRK is important and favored over BTIM but some BTIM added to the BRK is better than none. If unrestricted borrowing were allowed it would be foolish to choose the corresponding portfolio in Table 8. The various underlying assumptions are only approximations with varying degrees of validity: Stock prices do not change continuously ; portfolios can' t be adjusted continuously; transactions are not costless; the borrowing rate is greater than the Tbill rate; the after tax return, if different, needs to be used ; the process which generates securities returns is not stationary and our point estimates of the statistics in Table 7 are uncertain. We have also noted earlier that because "over The Kelly Criterion in Blackjack Sports Betting, and the Stock Market ( h <J: 821 The Kelly Criterion in Blackjack Sports Betting. and the Stock Market 417 Table 8 Optimal portfolio allocations with various assumptions about borrowing The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. Security fraction Security No borrowing 50% margin Berkshire BioTime S&P 500 Tbills .63 .37 .00 .00 1.50 .50 .00 1.00 12.61 19.04 .36 .29 .62 .45 2.10 2.03 Unrestricted borrowing 6.26 1.18 Portfolio growth rate Mean Standard deviation helling" is much more harmful than underbetting, "fractional Kelly" is prudent to the extent the results of the Kelly calculations reflect uncertainties. In fact, the data used comes from part of the period 19821997, the greatest bull market in history. We would expect returns in the future to regress towards the mean so Ihe means in Table 7 are likely to be overestimates of the near future. The data set is necessarily short, which introduces more uncertainty, because it is limited by the amount of BTIM data. As a sensitivity test, Quaife used conservative (mean, std. dev.) values for Ihe price relatives (not their logs) for BRK of (1.15, .20), BTIM of (1.15, 1.0) and the S&P 500 from 19261995 from Ibbotson (1998) of (1.125, .204) and the correlations from Table 7. The result was fractions of 1.65, .17, .18 and 1 .00 respectively for BRK, BTIM, S&P 500 and Tbills. The mean growth rate was . 19 and its standard deviation was .30. 8.3. The recommendation and the result The 50% margin portfolio reallocations of Table 8 were recommended to XYZ Corp.'s hoard on 8117/97 and could have been implemented at once. The boardelected to do nothing. On 10/9/97 (in hindsight, a good sale at a good price) it sold some BTIM and left the proceeds in cash (not good). Finally on 2/9/98 after a discussion with hoth Quaife and the author, it purchased 10 BRK (thereby gaining almost $140,000 hy 3/31/98, as it happened). The actual policy, led to an increase of 73.5%. What would have happened with the recommended policy with no rebalance and with one rebalance on 1O/6/97? The gains would have been 117.6% and 199.4%, respectively. The gains over the suboptimal board policy were an additional $475,935 and $1,359,826, respectively. The optimal policy displays three important features in this example: the use of leverage, the initial allocation of the portfolio, and possible rebalancing (reallocation) of the portfolio over time. Each of these was potentially important in ,determining the final 822 E. 0. Thorp E.O. The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. 418 T//(J/jJ result. The potential impact of continuously rebalancing to maintain maximum margin is illustrated in Thorp and Kassouf (1967), Appendix A, The Avalanche Effort. The large loss from the suboptimal policy was much more than what would have been expected because BRK and BTIM appreciated remarkably. In .62 years, BRK was up 60.4 % and BTIM was up 62.9%. This tells us thatatypicallyin the absence of rebalancing, the relative initial proportions of BRK and BTIM did not matter much over the actual time period. However, rebalancing to adjust the relative proportions of BRK and BTIM was important, as the actual policy 's sale of some BTIM on 10/9/97 illustrated. Also, rebalancing was important for adjusting the margin level as prices, in this instance, rose rapidly. Table S illustrates what we might have normally expected to gain by using 50% margin, rather than no margin. We expect the difference in the medians of the portfolio distributions to be $1,OSO,736[exp(.62 x .62)  exp(.36 x .62)] = $236,316 or 21.9% which is still large. 8.4. The theory for a portfolio of securities Consider first the unconstrained case with a riskless security (Tbills) with portfolio fraction fil and n securities with portfolio fractions fl , .. . , f". Suppose the rate of return on the riskless security is r and, to simplify the discussion, that this is also the rate for borrowing, lending, and the rate paid on short sale proceeds. Let C = (sii) be the matrix such that sij, i, j = I, ... , n, is the covariance of the i th and jth securities and M = (m I , /112, ... , 111/1)7 be the row vector such that 111; , i = I, ... , n, is the drift rate of the ith security. Then the portfolio satisfies fil 111 + ... + f/l = I, = f()r + fll11l + ... + f"I11/l = r + fdl11l  r) + ... + f;,(I11/l  r) =r+FT(MR), .1'2 = FTCF (S.I) where FT = (II , ... , f;,) and T means "transpose", and R is the column vector (r, r, ... , r) oflength n. Then our previous formulas and results for one security plus a riskless security apply to goo UI , ... , f;,) = m  .1'2/2. This is a standard quadratic maximization problem. Using (S.I) and solving the simultaneous equations (Jgoo/(Jfj = 0, i = 1, ... , n, we get ., F* = C  I[M  goout , ... , f;;) R], = r + (F*)TCF* /2 (S.2) where for a unique solution we require C I to exist, i.e., det C # O. When all the securities are uncorrelated, C is diagonal and we have I;* = (111;  r) /s;; or ft = (m;  r) /s;, which agrees with Equation (7.3) when II =, 1. The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 823 Ch. 9: 419 The Kelly Criterion in Blackjack Sports Betting, and the Stock Market Note: BRK issued a new class of common, ticker symbol BRKB, with the old common changing its symbol to BRKA. One share of BRKA can be converted to 30 shares of BRKB at any time, but not the reverse. BRKB has lesser voting rights and no right to assign a portion of the annual quota of charitable contributions. Both we and the market consider these differences insignificant and the A has consistently traded at about 30 times the price of the B. If the price ratio were always exactly 30 to 1 and both these securities were included in an analysis, they would each have the same covariances with other securities, so det C = and C  i does not exist. If there is an initial margin constraint of q , 0 ~ q ~ 1, then we have the additional restriction ° i!Ii + ... + ifni ~ l/q. (8 .3) The ndimensional subset in (8.3) is closed and bounded. If the rate for borrowing to finance the portfolio is rb = r + eb, e" :) 0, and the rate paid on the short sale proceeds is r ,,' = r  e,,', e" :) 0, then the m in Equation (8.1) is altered. Let x+ = max(x , 0) and x = max(O, x) so x = x +  x  for all x. Define f+ = ft + ... + t,;, the fraction of the portfolio held long. Let f = f) + .. . + t,;, the fraction of the portfolio held short. Case 1. f+ ~ I m =r+ !I(m) r) + ... + fll(m ll  r) es/ . (8.4.1 ) Case 2. f+ > 1 m =r+ !I(m)  r) + . .. + fll(m ll  r)  e,,(j+  1)  es/ . (8.4.2) 9. My experience with the Kelly approach How does the Kellyoptimal approach do in practice in the securities markets? In a littleknown paper (Thorp, 1971) I discussed the use of the Kelly criterion for portfolio management. Page 220 mentions that "On November 3, 1969, a private institutional investor decided to ... use the Kelly criterion to allocate its assets". This was actually a private limited partnership, specializing in convertible hedging, which I managed. A notable competitor at the time (see Institutional Investor (1998» was future Nobel prize winner Harry Markowitz. After 20 months, our record as cited was a gain of 39.9% versus a gain for the Dow Jones Industrial Average of +4.2%. Markowitz dropped out after a couple of years, but we liked our results and persisted. What would the future bring? Up to May 1998, twenty eight and a half years since the investment program began. The partnership and its continuations have compounded at approximately 20% annually with a standard deviation of about 6% and approximately zero correlation with the market ("market neutral"). Ten thousand dollars would, tax exempt, now be worth 18 824 E. O. Thorp 420 £.0. Thorp million dollars. To help persuade you that this may not be luck, 1 estimate that during this period I have made about $80 billion worth of purchases and sales ("action" , in casino language) for my investors. This breaks down into something like one and a quarter million individual "bets" averaging about $65,000 each , with on average hundreds of "positions" in place at anyone time. Over all , it would seem to be a moderately "long run " with a high probability that the excess performance is more than chance. The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. 10. Conclusion Those individuals or institutions who are long term compounders should consider the possibility of using the Kelly criterion to asymptotically maximize the expected compound growth rate of their wealth. Investors with less tolerance for intermediate term risk may prefer to use a lesser function. Long term compounders ought to avoid using a greater fraction ("overbetting"). Therefore, to the extent that future probabilities are uncertain, long term compounders should further limit their investment fraction enough to prevent a significant risk of overbetting. Acknowledgements T thank Dr. Jerry Baese\, Professor Sid Browne, Professor Peter Griffin, Dr. Art Quaife, and Don Schlesinger for comments and corrections and to Richard Reid for posting this chapter on his website. I am also indebted to Dr. Art Quaife for allowing me to use his analysis in the case study. This chapter has been revised and expanded since its presentation at the 10th International Conference on Gambling and Risk Taking. Appendix A. Integrals for deriving moments of E 00 10(a 2, /)2) = ( DO exp[  (a 2x 2 +b 2j x 2)]dx, Jo III(a 2, b2) = ( DO xllexp[  (a2x2+b2 j x2)]dx . Jo Given 10 find 12 lo(a 2, b 2) = roo exp[  (a 2x 2 +I}jx 2)]dX Jo =  f~ exp[ _(a 2 j u2 + h2u2)]( du j ll?) The Kelly Criterion in Blackjack Sports Betting, and the Stock Market I 'Ii , <J: 825 The Kelly Criterion in Blackjack Sports Belling, and the Stock Market = "here x l/u and dx 421 = du/u 2 so The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. hCllce (a L 2 b2 ) 2, 10 = 1 00 = I (b 2 0, a2 ) = 2y'li I b Ie2lab l' exp[ _(a 2x 2 + b 2 /x2)] dx where U = exp[·]' dV = dx, dU = = U . VI~  (exp[· ])(2a 2x 1 00 (A. I) V dU + 2b 2x  3 ) and V = x so 10 = exp[ _(a 2x 2 + b 2 /x2)] . xl~ 1 00 (2a 2x 2 + 2b 2/x2) exp[ _(a 2x 2 + b 2 /x2)] dx = 2a2h(a2, b 2 )  2b2L2(a2 , b 2 ). Hence: 10(a 2 , b 2 ) = 2a 2 h(a 2 , b 2 )  2b 2L2(a 2, b 2 ) and L2 (a 2 , b 2 ) = 10(b 2 , a 2 ) so substituting and solving for h gives Comments. ( I) We can solve for all even n by using 10 , 12 and h and integration by parts. (2) We can use the indefinite integral 10 corresponding to 10, and the previous methods, to solve for 12, h, and then for all even n. Since then Appendix B. Derivation of formula (3.1) This is based on a note from Howard Tucker. Any errors are mine,. 826 E. O. Thorp 422 £.0. Thorp From the paper by Paranjape and Park, if x(t) is standard Brownian motion, if a =I 0, b > 0, p(X(t) ~ at + b, ~ (aT + B  S)} 101  exp {  = 0 ~ t ~ T I X(T) = s) if s ~ aT + b, if s > aT + B. The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. Write this as: p(X(t) ~ at +b, 0 ~ t ~ T ~. 1  I X(T)) + b  X(T))~} ex p { 2b(aT if X(T) ~ aT + b. Taking expectations of both sides of the above, we get p(X(t) ~ at aT+I> = / + b, (1  0 ~ t ~ T) 1 e  2iJ(aT+bs)I / 1') _ _ e  s2/ 2T ds J2nT  00 = _1_ /aT + I> e· . 2/21" ds I J2n T 00 e  2ab " _ _ _ /"1' + 1> e (.1. 2b) 2/21 ds. J2n T 00 Hence + b during [0, T)) = 1  previous probability .2 /21" ds + e  2ab . _I_ /aT  I> e  u2, /21 du, P(X going above line at = _I_ JOO e I J2nT a1'+iJ where u = s  2b. J2nT  00 (B.l) Now, when a = 0, b > 0, P [ sup O~/ ~ T ] V(21°O ;;;r X(t) ~ b = nT 2 ' e V / 21 dv, I> which agrees with a known formula (see, e.g., p. 261 of Tucker (1967)). In the case a > 00, since fliT + 0 and f l = s.d. of X(T) , the first integral + 0, the second integral + 1, and P(X ever rises above line at + b) = e 2"I>. Similarly, in the case a < 0, peever rises above line at + b) = 1. The theorem it comes from is due to Sten Malmquist, On certain confidence contours for distribution functions, Ann. Math. Stat. 25 (1954), pp. 523533. This theorem is stated in S.R. Paranjape and C. Park, Distribution of the supremum of the twoparameter YehWiener process on the boundary, J. App!. Prob. 10 (1973). Letting ex = a fl, f3 = b I fl, formula (B . 1) becomes 0, when T + PO = N( ex  f3) + e 2afi N(ex  f3) where ex, f3 > 0 or The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 827 Ch. 9: 423 Th e Kelly Criterion in Blackjack Sports Betting, and the Stock Market p(x(t) :( at + b, 0 :( t :( T) = 1  PC) = N(a + fJ)  e  2afJ N(a  fJ) for the probability the line is never surpassed. This follows from: _1_1 00 e·\·2j2T ds = .J2rrT aT+b _1_1 00 ~ a./f+bj./f e x2j2 dx = N( a  fJ ) and The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. 1 j aTb ? e  u j2T du = N(a  fJ) .J2rrT  00 where s = aT + b, x = s / .JT = a.JT + b / .JT, a The formula becomes: + b)] ~ 0: 0 :( t :( T) fJ) + e 2a/J N(a  fJ) fJ) + e 2afJ N(a  fJ), a , fJ = a.JT and fJ = b / .JT. p(sup[X(t)  (at = N(a = N(a  > O. Observe that PC) < N(a  m+ N(a  fJ = ja a(x) dx + 1 00 fJ) = {I  N(a +fJ)} + N(a  fJ) a(x) dx < I a+fJ 00 as it should be. Appendix C. Expected time to reach goal Reference: Handbook of Mathematical Functions, Abramowitz and Stegun, Editors, N.B.S. Applied Math. Series 55, June 1964. P. 304, 7.4.33 gives with erf z == f~ e r2 dt the integral: Jrr f exp{ _(a 2 x 2 + b2/x2) }dx = :  [e 2a !J erf(ax + b/x) + e  211 !J erf(ax  b/x)] + C, a i O. (C.l) Now the left side is > 0 so for real a, we require a > 0 otherwise the right side is < 0, a contradiction. We also note that p. 302, 7.4.3. gives roo exp{ (at 2 + b/ t 2 )} dt = ~ ~e2~ 2V~ 10 with ma > 0, mb > O. (C.2) 828 E. 0. Thorp 424 E.O. Thorp To check (C.2) v. (C.l), suppose in (C.l) a > 0, b > 0 and find Iim x > o and limt> oo of erf(ax + b/x) and erf(ax  b/x), lim (ax x.j.o+ + b/x) = +00, lim (ax + b / x) = +00, ..\' 7 00 lim (ax  b/x) = 00, x.j.o+ lim (ax  b/x) = x +oo +00 . The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. Equation (C.l) becomes Jii e 2a l> [erf(oo) _ erf(oo)] = Jii e  2ab2erf(00) = Jii e 2ab 4a 4a 2a since we know erf( 00) = 1. In (C.2) replace a by a 2 , b by b 2 to get which is the same. Note: if we choose the lower limit of integration to be 0 in (C. I), then we can find C: 0+ 0= [ 10 Jii exp{ _(a 2x 2 + b 2/x2)} dx =  [e 2a l>erf(00) + e  2ah erf( 00)] + C 4a = Jii [e 2a l> _ e 211h ] 4a + c. Whence F(x) == foxexp{ _(a 2x 2 + b 2/ x2)} dx = Jii (e 2"j,[erf(ax 4a + b/x)  I] + e  211 j,[erf(ax  b/x) + I]}. (C.3) To see how (C.3) might have been discovered, differentiate: F'(x) = exp{(a 2x 2 +b 2/x2)} = Jii {e 211h (a _ b/x 2 ) erf'(ax 4a + e  211h (a Now erf' (z) = + b/x 2) erf'(ax  + b / x) b/x)}. Jrr exp(  Z2) so erf'(ax+b / x) 2 = 2 = Jiiexp[(ax+b/x)2] = 2 Jii e  2a/J exp{ _(a 2x 2 + b 2 /x2)} Jiiexp{(a2x2+b2/x2+2ab)} , The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 829 Ch. 9: 425 The Kelly Criterion in Blackjack Sports Betting, and the Stock Market and, setting b + b, whence {]rr(a  b/x 2) + ]rr(a F'(x) = : ~{2a} exp{ _(a 2x2 + b 2/x2)} The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. = 2a Case of interest: a < 0, b > O. Expect: + b/x2)} exp{ _(a 2x 2 + b 2/x2)} = exp{ _(a 2x 2 + b 2/x2)} . b > 0, a ::;; 0 ::::} F(T) t I as T + b> 0, a > 0 ::::} F(T) t c < 1 as T + 00. 00, If b > 0, a = 0: + N(fi) = 2N(b/.Jf) t Also, as expected F(T) t I as h ~ O. F(T) = N(f3) 2N(0) = 1 as T t 00. If b > 0, a < 0: See below. If b > 0, a > 0: F(T) = N(afl  h/fl) +e 2ab N(a.Jf b/.Jf) + N(oo) = e 2,,/! < +c 211i>N(00) I as T i 00. This is correct. If b .= 0: F(T) = N ((/ Jf) + N(a~) = l. This is correct. Let F(T) = P(X (I) ;? (/1 + h for some t, 0 ::;; t ::;; T) which equals N( a  f3) + e 2ab N(a  f3) where a = (/ Jf and fi = b/~ so ab = af3; we assume b > 0 and a < 0 in which case 0 ::;; F( T)' I and limT +00 F(T) = I, limT +0 F(T) = 0; F is a probability distribution funct illll: lim F(T) = N( (0) + c 2,,/, N (  00) = 0, T+O lim F(T)=N(+oo)+c 211"N(_00)=l. T+oo The density function is .nn = a a aT aT P'(T) =  (  a /l)N'(  a  f3) +e 211"_(a  f3)N'(a  f3) where il( y I ;IF 1 IIF 1/ 2 _Afi = _ _I_hl.. aT :2 l /2 The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. 830 E. O. Thorp 426 E.O. Thorp The expected time to the goal is Eoo = T 1 00 o I / 2 =2 x T  x dT= 2x dx be IIh 1 00 Tf(T) dT =   ' 1 ~ 2b e lIh   ~  T (a 2 T I / 2 exp { 2 0 1 {[ (~) 00 exp   a 2 x + b 2 /T) 2 l 2 (~) x +  ') } dT, ]}  2 dx Now I 0 (a 2 b2 ) = , jJi e 2111iJI 21al lo( (~r, (~r) ~ 2b e IIi, yJr Eoo =   ~e ~ ~Ial = so : ' I e I"h l  llIhl =  l ') lal ' whence a < 0, b > O. Note: 2 be IIh f(T)=F'(T)=  T  3 / 2 exp {  (a T ~ + b 2/ T) } > 0 2 for all a, e.g., a < 0, so F(T) is monotone increasing. Hence, since IimT .oo F(T) = 1 for a < 0 and limr .oo F(T) < I for a > 0,0 :::;; F(T) :::;; I for all T so we have more The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 831 Ch. 9: 427 Th e Kelly Criterion in Blackjack Sports Belling, and the Stock Market confidence in using the formula for a < 0 too. Eoo(a , b) I 0 as I Check: The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. Eoo(a, b) t as b E oo (a , b) t aslal I lim E oo (a, b) = note 00 t yes , yes, yes, as suspected. +00 a~O+ This leads us to believe that in a fair coin toss (fair means no drift) and a gambler with finite capital, the expected time to ruin is infinite. This is correct. Feller gives D = z(a  z) as the duration of the game, where z is initial capital, ruin is at 0, and (/ is the goal. Then lim,, >oo D(a) = +00. Note: Eoo = b/lal means the expected time is the same as the point where aT + b crosses X(t) = O. See Figure 2. E oo = h/lal , A = C/ Xo = _111 /.1'" , (/ = normali/.ed goal. + fl + £I IIl( I pq{ln[(l + fl / ( 1  m = p In(1 s2 = h = InA,  n == g(f), n1l 2 , Kelly fraction f* = p  £f . Form> 0, g(f*) = p In2p + q In2q, E oo = (InA).I'2/ g (f). Now this is the expected time in variance units . However s2 variance units = 1 trial so Eoo n(A , f) InA InA == ~ = g(f) = ;;; is the expected number of trials. Chcck: I/(A , f) II t (A , f) + t II (A , f) II(A, f) + as A 00 t, as A + 00, as m I 0, 00 as m + O. N(I\\ t:1 I ) it;IS uilique maximum at g(f*) where f* = p  q, the "Kelly fraction", thcn'l, II,' 1111 , I) has a unique minimum for f = f*, Hence f* reaches a fixed goal in leas I ("I"',It'lIIIlIlC in this, the continuous case, so we must be asymptotically close to l e a~1 ( " IIt" 1111lL' ill the discrete case, which this approximates increasing by well in thl' "('II',,' .. I I h,' ( 'I T (Central Limit Theorem) and its special case, the normal approxill);11 It III I .. 1111' hlllol11ial distribution, The difference here is the trials are asymmetric. Till' 1"1',111\ " ,lIltllll'galive step sizes are unequal. !t." 832 428 E. 0. Thorp £.0. Thorp The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. References Associates First Capital Corporation, 1998. 1997 Annual Report. Associates First Capital Corporation, Dallas, TX. Black, E, Scholes, M., 1973. The pricing of options and corporate liabilities. Journal of Political Economy 81, 637659. Breiman, L., 1961 . Optimal gambling systems for favorable games. In: Fourth Berkeley Symposium on Probability and Statistics, vol. 1, pp. 6578. Browne, S., 1996. Reaching Goals by a Deadline: ContinuousTime Active Portfolio Management. Columbia University, New York. Browne, S. , 1997. The return on investment from proportional investment strategies. Advances in Applied Probability 30 (I), 216238. Buffett. M. , Clark. D. , 1997. Buffettology. Rawson Associates, Simon and Schuster, New York. Cox, D.R., Miller, H.D., 1965. The Theory of Stochastic Processes. Wiley, New York. Feller, w., 1966. An Introduction to Probability Theory and Its Applications, vol. I, Revised. Wiley, New York. Griffin, P.A., 1979. The Theory of Blackjack. Huntington Press, Las Vegas. Revised 1995. Hagstrom, R.G. Jr. , 1994. The Warren Buffett Way. Wiley, New York. Hagstrom, R.G. Jr., 2004. The Warren Buffett Way, second ed. Wiley, New York. Hecht, R. , 1995. Private correspondence. Ibbotson Associates, 1998. Yearbook: Stocks, Bonds, Bills and Inflation (or any later edition). Ibbotson Associates, Chicago. Institutional Investor, 1998. Ivory Tower Investing, pp. 4355 (see p. 44), March. Kelly, J.L. , 1956. A new interpretation of information rate. Bell System Technical Journal 35, 917926. Kilpatrick, A. , 1994. Of Permanent Value, the Story of Warren Buffet. Distributed by Southern Publishers Group, Birmingham, AL. Kritzman, M., 1998. Risk and utility: basics. In: Bernstein, Damodaran (Eds.), Investment Management. Wiley, New York. Chapter 2. Lowenstein, R., 1995. The Making of an American Capitalist. Random House, New York. Markowitz, H., 1959. Portfolio Selection. Cowles Monograph, vol. 16. Wiley, New York. McEnally, R.W., 1986. Latane's bequest: the best of portfolio strategies. Journal of Portfolio Management 12 (2),2130, Winter. Quaife, A., 1995. Using the Sharpe ratio to evaluate investments. The Trans Times 4 (1) , February. Trans Time Inc., Oakland, CA. Rotando, L.M., Thorp, E.O., 1992. The Kelly criterion and the st