AIAR Q3 2016 05 KellyCapital

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AIAR Q3 2016 05 KellyCapital
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  Understanding the Kelly Capital Growth Investment Strategy Dr William T. Ziemba Alumni Professor Sauder School of Business University of British Columbia Investment Strategies 49 Using Kelly Capital Growth Strategy Introduction to the Kelly Capital Growth Criterion and Samuelson’s Objections to it Te Kelly capital growth criterion, which maximizes the expected log o final wealth, provides the strategy that maximizes long run wealth growth asymptotically or repeated investments over time. However, one drawback is ound in its very risky behavior due to the log’s essentially zero risk aversion; consequently it tends to suggest large concentrated investments or bets that can lead to high  volatility in the short-term. Many investors, hedge unds, and sports bettors use the criterion and its seminal application is to a long sequence o avorable investment situations.Edward Torp was the first person to employ the Kelly Criterion, or “ Fortune’s Formula”   as he called it, to the game o blackjack. He outlines the process in his 1960 book Beat the Dealer   and his findings changed the way this game was played once he had demonstrated that there was a winning strategy. As applied to finance, a number o note-worthy investors use Kelly strategies in various orms, including Jim Simons o the Renaissance Medallion hedge und. Te purpose o this paper is to explain the Kelly criterion approach to investing through theory and actual investment practice. Te approach is normative and relies on the optimality properties o Kelly investing. Tere are, o course, other approaches to stock and dynamic investing. Besides mean-variance and its extensions there are several important dynamic theories. Many o these are surveyed in MacLean and Ziemba (2013). An interesting continuous time theory based on descriptive rather than normative concepts with arbitrage and other applications is the stochastic portolio theory o Fernholz and colleagues, see or example, Fernholz and Shay(1982), Fernholz (2002), and Karatzas and Fernholz (2008). Tey consider the long run perormance o portolios using specific distributions o returns such as lognormal. Te Kelly approach uses a specific  Investment Strategies Alternative Investment Analyst Review Quarter 3 ã 2016 50 utility unction, namely log, with general asset distributions. What is the Kelly Strategy and what are its main properties? Until Daniel Bernoulli’s 1738 paper, the linear utility o wealth was used, so the value in ducats would equal the number o ducats one had. Bernoulli postulated that the additional value was less and less as wealth increased and was, in act, proportional to the reciprocal o wealth so,where u  is the utility unction o wealth w , and primes denote differentiation. Tus concave log utility was invented.In the theory o optimal investment over time, it is not quadratic (one o the utility unction behind the Sharpe ratio), but log that yields the most long-term growth asymptotically. Following with an assessment o that aspect, the Arrow-Pratt risk aversion index or log( w ) is:which is essentially zero. Hence, in the short run, log can be an exceedingly risky utility unction with wide swings in wealth  values.John Kelly (1956) working at Bell Labs with inormation theorist Claude Shannon showed that or Bernoulli trials, that is win or lose 1 with probabilities  p  and q  or  p+q= 1 , the long run growth rate, G , namely where t   is discrete time and w 1  is the wealth at time t   with w 0  the initial wealth is equivalent to max E [log w ] Since w t   = (1+ƒ)  M   (1 –ƒ) t-M    is the wealth afer t   discrete periods,  ƒ is the raction o wealth bet in each period and M o the t   trials are winners.Ten, substituting W  t   into G  givesand by the strong law o large numbersTus the criterion o maximizing the long run exponential rate o asset growth is equivalent to maximizing the one period expected logarithm o wealth. So an optimal policy is myopic in the sense that the optimal investments do not depend on the past or the uture. Sincethe optimal raction to bet is the edge  ƒ * =  p – q . Te edge is the expected value or a bet o one less the one bet. Tese bets can be large. For example, i  p= 0.99  and q= .01 , then  ƒ * = 0.98 , that is 98% o one’s ortune. Some real examples o very large and very small bets appear later in the paper. I the payoff odds are +B or a win and -1 or a loss, then the edge is Bp – q  andSo the size o the investments depend more on the odds, that is to say, the probability o losing, rather than the mean advantage. Kelly bets are usually large and the more attractive the wager, the larger the bet. For example, in the trading on the January turn-o-the-year effect with a huge advantage, ull Kelly bets approach 75% o initial wealth. Hence, Clark and Ziemba (1988) suggested a 25% ractional Kelly strategy or their trades, as discussed later in this article.Latane (1959, 1978) introduced log utility as an investment criterion to the finance world independent o Kelly’s work. Focusing, like Kelly, on simple intuitive versions o the expected log criteria, he suggested that it had superior long run properties. Chopra and Ziemba (1993) have shown that in standard mean- variance investment models, accurate mean estimates are about twenty times more important than covariance estimates and ten times variances estimates in certainty equivalent value. But this is risk aversion dependent with the importance o the errors becoming larger or low risk aversion utility unctions. Hence, or log w  with minimal risk aversion, the impact o these errors is o the order 100:3:1. So bettors who use E log  to make decisions can easily over bet. Leo Breiman (1961), ollowing his earlier intuitive paper Breiman (1960), established the basic mathematical properties o the expected log criterion in a rigorous ashion. He proved three basic asymptotic results in a general discrete time setting with intertemporally independent assets.Suppose in each period, N  , there are K   investment opportunities with returns per unit investe 1 , ,  K   N N   X X   . Let 1 ( , , )  K  Λ = Λ Λ   be the raction o wealth invested in each asset. Te wealth at the end o period N   isIn each time period, two portolio managers have the same amily o investment opportunities,  X  , and one uses a Λ  which maximizes E log w N   whereas the other uses an essentially different   strategy, Λ , so they differ infinitely ofen, that is, TenSo the wealth exceeds that with any other strategy by more and more as the horizon becomes more distant. Tis generalizes the Kelly Bernoulli trial setting to intertemporally independent and stationary returns.Te expected time to reach a preassigned goal A is asymptotically least as A increases with a strategy maximizing E log w N . Assuming a fixed opportunity set, there is a fixed raction strategy that maximizes E log w N  , which is independent o N  . ( ) 1/ or ( ) log( ) u w w u w w ′ = = 1( )  A u R wu w ′′= − =′ 1/0 limlog t t t  wGw →∞  =    lim log(1 ) log(1 ) log(1 ) log(1 ) t   M t M G f f p f q f  t t  →∞ − = + + − + + + −   [ ] log G E w = max ( ) log(1 ) log(1 ) G f p f q f   = + + −  Bp q edge f   B odds ∗  −= = 11 .  K  N i Ni N i w X w −=  = Λ   ∑ log log ( ) .  N N   E w E w ∗ Λ − Λ →∞ ( ) ( ) * lim  N  N  N  ww →∞ Λ→∞Λ  51 Using Kelly Capital Growth Strategy Consider the example described in Exhibit 1. Tere are five possible investments and i we bet on any o them, we always have a 14% advantage. Te difference between them is that some have a higher chance o winning than others. For the latter, we receive higher odds i we win than or the ormer. But we always receive 1.14 or each 1 bet on average. Hence we have a avorable game. Te optimal expected log utility bet with one asset (here we either win or lose the bet) equals the edge divided by the odds. So or the 1-1 odds bet, the wager is 14% o ones ortune and at 5-1 its only 2.8%. We bet more when the chance that we will lose our bet is smaller. Also, we bet more when the edge is higher. Te bet is linear in the edge so doubling the edge doubles the optimal bet. However, the bet is non-linear in the chance o losing our money, which is reinvested so the size o the wager depends more on the chance o losing and less on the edge.Te simulation results shown in Exhibit 2 assume that the investor’s initial wealth is $1,000 and that there are 700 investment decision points. Te simulation was repeated 1,000 times. Te numbers in Exhibit 2 are the number o times out o the possible 1,000 that each particular goal was reached. Te first line is with log or Kelly betting. Te second line is hal Kelly betting. Tat is, you compute the optimal Kelly wager but then blend it 50-50 with cash. For lognormal investments α -ractional Kelly wagers are equivalent to the optimal bet obtained rom using the concave risk averse, negative power utility unction, –w –β , where 11  β  α  − = . For non lognormal assets this is an approximation (see MacLean, Ziemba and Li, 2005 and Torp, 2010, 2011). For hal Kelly ( α =1/2) ,  β=–1  and the utility unction is - w -1 = –1/w . Here the marginal increase in wealth drops off as w 2 , which is more conservative than log’s w  . Log utility is the case  β→–∞, α =1  and cash is  β→–∞, α =0 .A major advantage o ull Kelly log utility betting is the 166 in the last column. In ully 16.6% o the 1,000 cases in the simulation, the final wealth is more than 100 times as much as the initial wealth. Also in 302 cases, the final wealth is more than 50 times the initial wealth. Tis huge growth in final wealth or log is not shared by the hal Kelly strategies, which have only 1 and 30, respectively, or these 50 and 100 times growth levels. Indeed, log provides an enormous growth rate but at a price, namely a very high volatility o wealth levels. Tat is, the final wealth is very likely to be higher than with other strategies, but the wealth path will likely be very bumpy. Te maximum, mean, and median statistics in Exhibit 2 illustrate the enormous gains that log utility strategies usually provide.Let us now ocus on bad outcomes. Te first column provides the ollowing remarkable act: one can make 700 independent bets o which the chance o winning each one is at least 19% and usually is much more, having a 14% advantage on each bet and still turn $1,000 into $18, a loss o more than 98%. Even with hal Kelly, the minimum return over the 1,000 simulations was $145, a loss o 85.5%. Hal Kelly has a 99% chance o not losing more than hal the wealth versus only 91.6% or Kelly. Te chance o not being ahead is almost three times as large or ull versus hal Kelly. Hence to protect ourselves rom bad scenario outcomes, we need to lower our bets and diversiy across many independent investments.Exhibit 3 shows the highest and lowest final wealth trajectories or ull, 34  , 12  , 14  and 18  Kelly strategies or this example. Most o the gain is in the final 100 o the 700 decision points. Even with these maximum graphs, there is much volatility in the final wealth with the amount o volatility generally higher with higher Kelly ractions. Indeed with 34  Kelly, there were losses rom about decision points 610 to 670.Te final wealth levels are much higher on average, the higher the Kelly raction. As you approach ull Kelly, the typical final wealth escalates dramatically. Tis is shown also in the maximum wealth levels in Exhibit 4. Probability    o Winning Odds Probability o Being Chosenin the Simulation at atEach Decision PointOptimal Kelly Bets Fractiono Current Wealth 0.571-10.10.140.382-10.30.070.2853-10.30.0470.2284-10.20.0350.195-10.10.028 Exhibit 1: Te Investments Source: Ziemba and Hausch (1986) Exhibit 2: Statistics of the Simulation Source: Ziemba and Hausch (1986) Final WealthStrategyMin Max Mean Median Number of times the nal wealth out of 1000 trials was > 500 > 1000 > 10,000 > 50,000 > 100,000 Kelly 18483,88348,13517,269916870598302166 Half Kelly 145111,77013,0698,043990954480301  Investment Strategies Alternative Investment Analyst Review Quarter 3 ã 2016 52 Tere is a chance o loss (final wealth is less than the initial $1,000) in all cases, even with 700 independent bets each with an edge o 14%.I capital is infinitely divisible and there is no leveraging, then the Kelly bettor cannot go bankrupt since one never bets everything (unless the probability o losing anything at all is zero and the probability o winning is positive). I capital is discrete, then presumably Kelly bets are rounded down to avoid overbetting, in which case, at least one unit is never bet. Hence, the worst case with Kelly is to be reduced to one unit, at which point betting stops. Since ractional Kelly bets less, the result ollows or all such strategies. For levered wagers, that is, betting more than one’s wealth with borrowed money, the investor can lose much more than their initial wealth and become bankrupt. Selected Applications In this section, I ocus on various applications o Kelly investing starting with an application o mine. Tis involves trading the turn-o-the-year effect using utures in the stock market. Te first paper on that was Clark and Ziemba (1988) and because o the huge advantage at the time suggested a large ull Kelly wager approaching 75% o initial wealth. However, there are risks, transaction costs, margin requirements, and other uncertainties which suggested a lower wager o 25% Kelly. Tey traded successully or the years 1982/83 to 1986/87 - the first our years o utures in the OY; utures in the S&P500 having just begun at that time. I then continued this trade o long small cap minus short large cap measured by the Value Line small cap index and the large cap S&P500 index or ten more years with gains each Exhibit 3: Final Wealth rajectories: Ziemba-Hausch (1986) Model. Source: MacLean, Torp, Zhao and Ziemba (2011) Exhibit 4: Final Wealth Statistics by Kelly Fraction: Ziemba-Hausch (1986) Model Kelly Fraction Source: MacLean, Torp, Zhao and Ziemba (2011) b) Lowesta) Highest Statistic 1.0k 0.75k 0.50k 0.25k 0.125k Max 318854673 4370619 1117424 27067 6330Mean 524195 70991 19005 4339 2072Min 4 56 111 513 587St. Dev. 8033178 242313 41289 2951 650Skewness 35 11 13 2 1Kurtosis 1299 155 278 9 2 > 5 × 10 1981 2000 2000 2000 200010 2 1965 1996 2000 2000 2000 > 5 × 10 2 1854 1936 1985 2000 2000 > 10 3 1752 1855 1930 1957 1978 > 10 4 1175 1185 912 104 0 > 10 5 479 284 50 0 0 > 10⁶ 111 17 1 0 0
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