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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 portolio theory o Fernholz and colleagues, see or example, Fernholz and Shay(1982), Fernholz (2002), and Karatzas and Fernholz (2008). Tey consider the long run perormance o portolios 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 inormation 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 portolio 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 diversiy 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 successully 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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