PDF Good and bad properties of the Kelly criterion

[Pages:11]Good and bad properties of the Kelly criterion

Leonard C. MacLean, Herbert Lamb Chair (Emeritus),School of Business,

Dalhousie University, Halifax, NS

Edward O. Thorp, E.O. Thorp and Associates, Newport Beach, CA Professor Emeritus, University of California, Irvine

William T. Ziemba, Professor Emeritus, University of British Columbia, Vancouver, BC Visiting Professor, Mathematical Institute, Oxford University, UK

ICMA Centre, University of Reading, UK University of Bergamo, Italy

January 1, 2010

Abstract We summarize what we regard as the good and bad properties of the Kelly criterion and its variants. Additional properties are discussed as observations.

The main advantage of the Kelly criterion, which maximizes the expected value of the logarithm of wealth period by period, is that it maximizes the limiting exponential growth rate of wealth. The main disadvantage of the Kelly criterion is that its suggested wagers may be very large. Hence, the Kelly criterion can be very risky in the short term.

In the one asset two valued payoff case, the optimal Kelly wager is the edge (expected return) divided by the odds. Chopra and Ziemba (1993), reprinted in Section 2 of this volume, following earlier studies by Kallberg and Ziemba (1981, 1984) showed for any asset allocation problem that the mean is much more important than the variances and co-variances. Errors in means versus errors in variances were about 20:2:1 in importance as measured by the cash equivalent value of final wealth. Table 1 and Figure 1 show this and illustrate that the relative importance depends on the degree of risk aversion. The

Special thanks go to Tom Cover and John Mulvey for helpful comments on an earlier draft of this paper.

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Good and bad properties

MacLean, Thorp, Ziemba

lower is the Arrow-Pratt risk aversion, RA = -u (w)/u (w), the higher are the relative errors from incorrect means. Chopra (1993) further shows that portfolio turnover is larger for errors in means than for variances and for co-variances but the degree of difference in the size of the errors is much less than the performance as shown in Figure 2.

Table 1: Average Ratio of Certainty Equivalent Loss for Errors in Means, Variances and

Covariances. Source: Chopra and Ziemba (1993)

Errors in Means Errors in Means Errors in Variances

Risk Tolerance* vs Covariances vs Variances

vs Covariances

25

5.38

3.22

1.67

50

22.50

10.98

2.05

75

56.84

21.42

2.68

20

10

2

Error Mean

Error Var

Error Covar

20

2

1

Nav Index

S&P500 0D3ec-04Mar-0Ju4n-04Sep-0D4ec-0M4 ar-0Ju5n-05Sep-0D5ec-05Dec-06

Date

Risk

tolerance=RT (w) =

100

1 2

RA

(w)

where

RA(w)

=

-

u u

(w) (w)

% Cash Equivalent Loss

11 10

9 8 7

6

5 4

3

2 1

0

0

0.05

0.10

0.15

Magnitude of error (k)

Means

Variances Covariances 0.20

Price (U.S. dollars)

15 14 13 12 11 10

9 8 7

6 5 4 3 2

1 0

11/68

10/73

Figure 1: Mean Percentage Cash Equivalent Loss Due to Errors in Inputs.

Since log has RA(w) = 1/w, which is close to zero, The Kelly bets may be exceedingly large and risky for favorable bets. In MacLean, Thorp, Zhao and Ziemba (2009) in this section of this volume, we present simulations of medium term Kelly, fractional Kelly and proportional betting strategies. The results show that with favorable investment opportunities, Kelly bettors attain large final wealth most of the time. But, because a long sequence of bad scenario outcomes is possible, any strategy can lose substantially even if there are many independent investment opportunities and the chance of losing at each investment

PNP S&P 500 U.S. T-Bill

11/77

10

2

Magnitude of Error (k)

Means

Variances

Covariances

Source: Based on data from Chopra and Ziemba (1993).

Good aFnidgubraed1p.8r.opeArtvieersage Turnover for Different Percentage CMhaancgLeesanin, TMheoarnps,, Ziemba Variances, and Covariances

Average Turnover (% per month) 50

40

Means

30 Variances

20

10

Covariances

10

5

10

15

20

25

30

35

40

45

50

Change (%)

Source: Based on data from Chopra (1993).

Figure 2: Average turnover for different percentage changes in means, variances and co-

variances. Source: Based on data from Chopra (1993)

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?2003, The Research Foundation of AIMRTM

decision point is small. The Kelly and fractional Kelly rules, like all other rules, are never

a sure way of winning for a finite sequence.

In Section 6 of this volume, we describe the use of the Kelly criterion in many applications and by many great investors. Two of them, Keynes and Buffett, were long term investors whose wealth paths were quite rocky but with good long term outcomes. Our analyses suggest that Buffett seems to act similar to a fully Kelly bettor (subject to the constraint of no borrowing) and Keynes like a 80% Kelly bettor with a negative power utility function -w-0.25, see Ziemba (2003). See the wealth graphs reprinted in section 6 from Ziemba (2005).

Graphs such as Figure 3 show that growth is traded off for security with the use of fractional Kelly strategies and negative power utility functions. Log maximizes the long run growth rate. Utility functions such as positive power that bet more than Kelly have more risk and lower growth. One of the properties shown below that is illustrated in the graph is that for processes which are well approximated by continuous time, the growth rate becomes zero plus the risk free rate when one bets exactly twice the Kelly wager.

Hence it never pays to bet more than the Kelly strategy because then risk increases (lower security) and growth decreases, so Kelly dominates all these strategies in geometric riskreturn or mean-variance space. See Ziemba (2009) in this volume.

As you exceed the Kelly bets more and more, risk increases and long term growth falls, eventually becoming more and more negative. Long Term Capital is one of many real

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Good and bad properties

MacLean, Thorp, Ziemba

Figure 2: Probability of doubling and quadrupling before halving and relative growth rates versus

fraction of wealth wagered for Blackjack (2% advantage, p=0.51 and q=0.49).

Source: McLean and Ziemba (1999)

Figure 3: Probability of doubling and quadrupling before halving and relative growth

rates versus fraction of wealth wagered for Blackjack (2% advantage, p=0.51 and q=0.49).

Source: MacLean and ZiemTbaable(13:9G9r9o)wth Rates Versus Probability of Doubling Before Halving for Blackjack

Kelly Fraction

P[Doubling before

of Current

Halving]

world instances in which overbetting led to dWiseaasltther. See Ziemba and Ziemba (2007) for

additional examples.

0.1

|

0.2

0.999 0.998

Thus long term growth maximRainzgine g in|vestors sh0.o3uld bet _Kelly or l0e.s9s8. We call be_tting

less

than

Kelly

"fractional

KBellalfcyok,rj"ackwhi|ch

is

0.4

sim0p.5ly

a

bleSnafder of

Kelly00..a98n49 d

cash. LeCssoGnrsoiwdethr

the negative power utility funTcetaimons | for ................
................

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