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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.16 No.3 pp.392-399
DOI : https://doi.org/10.7232/iems.2017.16.3.392

A Comparison of the Kelly Criterion and a Mean-Variance Model to Portfolio Selection with KOSPI 200

Gyutai Kim, Jong-Ho Shin*
Department of Industrial Engineering, Chosun University, Republic of Korea
Corresponding Author, jhshin@chosun.ac.kr
February 27, 2017 May 11, 2017

ABSTRACT

The Markowitz’s mean-variance model has been extensively discussed and widely used to solve an investment portfolio optimization problem. Like the model, the Kelly criterion has been also used for the same purpose. But the latter has received much less attention than the former from the academic and financial world. Now, it is strongly believed that the Kelly criterion provides investors with an investment performance better than or equal to the Markowitz’s model. In this paper, we constructed the portfolio optimization models corresponding to each of two models and made a comparison of their results to investigate which model would lead investors to achieve a better investment performance. To this end, we carried out a case study with KOSPI 200. As one of the findings derived from of the case study, we found that the Kelly criterion model generated much higher return and risk, and a less diversified portfolio than the Markowitz’s model.


초록


    1.INTRODUCTION

    An investment portfolio optimization is a critical subject in equity portfolio management, financial planning, and asset allocation. Its purpose is to maximize a rate of return on investment while minimizing the adverse risk inherent in the investment. Over the last five and half decades have a variety of mathematically complicated investment portfolio optimization models been proposed for the purpose to be fullfilled. Of these models, an arithmetic mean-variance (AM-V) model equivalent to the Markowitz’s mean-variance model, a geometric meanvariance model (GM-V) equivalent to the Kelly criterion model, and an expected utility model have, relatively speaking, received a considerable attention from researchers. Just for the convenient writing purpose, we will exchangeably use AM-V model for the Markowitz portfolio, and GM-V model for the Kelly criterion in the following sections.

    According to Markowitz (1952, 1991, 2000), the AM-V model begins with information regarding to individual investment assets and ends with results with respect to portfolios in general, and all information and results discussed in association with the model are assumed to stochastically behave in the context. He recognized risk of investment portfolios during the beginning of the 1950s and quantified it in variance. With the recognition and quantification of the risk, he introduced the notion of the AM-V efficient portfolio. Since then, it has played a focal role in financial markets as the investment decision rule of building an optimal investment portfolio by which the investors logically allocated their investment capital and rationalized the value of diversification. It implies that the investment capital on hand may be optimally allocated to each stock in the investment portfolio to acquire the maximum rate of return for the given conditions. To determine the optimal portfolio solution, in general, a quadratic programming problem has been set up and solved. But, Broadie (1997) proposed a linear programming technique to solve a nonlinear form of the Markowitz’s portfolio problem in 1997. Among many, Michaud (1989) describes pros and cons of the AM-V optimizers in detail.

    Another model, in contrast with the Markowitz’s AM-V model, is one that is formulated with the Kelly criterion. Kelly (1956) proposed his unique model, called “the Kelly criterion,” which was developed based on a geometric mean concept in 1956, leading geometric mean-variance criteria. Thorp (1971, 1997, 2008), Ziemba (1985), etc. have comprehensively studied and practiced the criterion in financial markets and gambiling games as well. In particular, Thorp (1971, 1997) has frequently explored in his writings and seminars that he realized a high return on his investment by adopting the criterion concept. In addition, he also argues that the Kelly criterion should take a place of a Markowitz’s portfolio theory for an investment portfolio construction. Brieman (1961) proved that the Kelly criterion asymptotically minimized time to accomplish a certain level of wealth and maximized wealth within the least period of a time horizon. Filkelstein and Whitley (1981) extended the Kelly’s and Brieman’s results, and demonstrated that the investors who made investment with the Kelly criterion gained a rate of return, on average, superior to those who did with another one. By the same token, Ziemba and Hausch (1985) also recognized that the Kelly criterion generated pretty much higher expected capital growth than any other investment strategy in case which investment was made over a long period of time.

    A number of studies have compared and presented the performances of the AM-V and GM-V models. Hakansson (1971) showed that the AM-V model led to quite different investment policies from the GM-V model. In this study, he found with a simple example involving two risky assets that i) the optimal GM-V portfolio was not even adjacent to the AM-V efficient frontier, ii) the only feasible portfolios leading to ruin in the long run were the AM-V efficient frontier, and iii) in the long run, the worst portfolio from a AM-V point of view generated better performance than most of the portfolios on the AM-V efficient frontier. Zhang (2010) stated more specifically that the optimal GM-V portfolio could lie either on or nearby to the AM-V efficient frontier. For more information regarding to the comparison, readers may refer to Estrada (2010), Fama and MacBeth (1974), Grauer (1978, 1981), Hakansson (1971), Hunt (2005a, 2005b), Roll (1973), Zhang (2010) , and so on. In this paper, taking the viewpoints of the Kelly criterion into consideration, we investigated the results of the Markowitz’s and Kelly portfolio model using the stock prices listed in KOSPI 200, and discussed them to provide a useful information for the investors.

    The remaining part of the paper is as follows: Since the Kelly criteria is unfamiliar with many readers, its fundamental concept is described in Section 2. In addition, Section 2 describes the basic motivation for which the GM-V efficient portfolio frontier dominates the AM-V model. Section 3 is concerned with discussing the results of two different portfolio optimization problems, which were executed with the real data of KOSPI 200. Finally, we will present a concluding remarks on the research from the portfolio management and decision-making perspectives.

    2.THE FUNDAMENTAL CONCEPT OF THE KELLY CRITERION

    In this section, we will introduce a fundamental concept of the Kelly criterion. Daniel Bernoulli (1954) first introduced a logarithmic utility function to value a risky game which is now well known as the St. Petersburg Paradox and stipulated that a marginal value of one unit of increase in wealth became less worthy as wealth increased. This stipulation can be mathematically expressed as follows (Thorp, 1971; Zhang, 2010):

    u ( x ) = 1 x
    (1)

    Equation (1) means that u(x) = log(x), where u(⋅) is a utility function, u′ is the first differentiation of the function, and x is a size of wealth or investment capital.

    The coefficient of absolute risk aversion of a logarithmic utility function is defined as

    A ( x ) = u ( x ) u ( x ) = 1 x
    (2)

    By examining Equation (2), we can recognize that it virtually approaches zero as wealth or investment capital increases. Thuis, it can be inferred that the logarithmic utility function is an exceedingly risky utility function in which the value of wealth greatly fluctuates in the short run. Motivated by the characteristics of the logarithmic utility function and Shannon’s information theory (Shannon, 1948), John Kelly proposed his criterion as previously mentioned that provides us with a way to maximize the long run growth rate of wealth or investment capital. However, John Kelly did not mention the logarithmic utility function in his original paper. It has been called in a number of different names such as the “geometric mean maximizing portfolio strategy,” “maximizing logarithmic utility,” “the capital growth criterion,” “the growthoptimal strategy,” and so on. In the following subsection, we will go for more detail about the criterion.

    2.1.More about the Kelly Criterion

    The discussion in this subsection is mainly based on Brieman’s and Thorp’s work (Thorp, 1969; Brieman, 1961). First of all, it is good to know the following Equation (3) to understand the derivation process of the Kelly criterion in advance.

    e n ln [ X N X 0 ] 1 / N = X N X 0
    (3)

    Here, X0 is investors’ initial capital(wealth) and XN is their wealth accumulated after N trials of investment have been made. There are several other notations we need to know. The term of f is generally defined as a betting ratio which is applied to the investors’ current wealth where 0 ≤ f < 1. W and L denote a number of successes and failures of the trials, respectively, for N = W + L.

    Also, we need to understand and keep in mind the following assumptions so that we may successfully implement the Kelly criterion (Kelly, 1956; Thorp, 1971, 1997; Zhang, 2010; Ziemba and Ziemba, 2007). Those are:

    • The expected rate of return must be greater than 0.

    • It must be possible to reinvest the cash flows earned from investment.

    • There must exist an independent relationship among the final cash flows.

    • The cash flows must follow a Bernoulli process.

    • The investing activities must be performed over a long period of time.

    • It must be possible to divide an investment capital infinitesimally.

    • The winning probability of p is constant and known to the investors.

    With the assumptions and notations defined above in mind, after N trials of investment, the investors’ wealth becomes

    X N = X 0 ( 1 f ) W ( 1 f ) L
    (4)

    Comibining Equataion (3) and (4) provides a general mathematical formula to express a logarithmic growth rate of the investors’ wealth as Equation (5).

    G ( f ) = ln ( X N / X 0 ) = W N ln ( 1 + f ) + L N ln ( 1 f )
    (5)

    In Equation (5), the term of G represents a logarithmic growth rate of return on investment. And its expected value, denoted by g(f) , is given by Equation (6).

    g ( f ) = E { ln [ X N X 0 ] 1 / N } = E { W N ln ( 1 + f ) + L N ln ( 1 f ) } = p ln ( 1 + f ) + q ln ( 1 f )
    (6)

    where p and q are a probability of successes and failures, respectively. More specifically stating, The term of g(f) means the expected long-term growth rate of return on the investment.

    To find an optimal value of f which maximizes g(f), we need to take the first and second partial differentiation of Equation (6) with respect to f, and do some of mathematical labor-work. Then, we may obtain

    g ( f ) = p ( 1 + f ) q ( 1 f ) = p q f ( 1 + f ) ( 1 f ) = 0
    (7)

    in which f* = pq = 2p − 1.

    In Equation (7), if p < 1 2 , then f* < 0 , which implies that a short selling is suggested. Since we do not restrict on the condition of p + q = 1, p, in Equation (7) may take value less than 0. However, if 1 2 < p < 1 , then there exists the non-trivial investment strategy of 0 < f* < 1.

    And now we may have

    g f = p ( 1 + f ) 2 q ( 1 f ) 2 < 0
    (8)

    Equation (8) says that g′(f) is monotone strictly decreasing on [0,1). Since g′(0) = pq = 2p − 1 and lim f 1 g f = , g(f) provides a unique maximum at f = f*, where g(f*) = plnp + qlnq + ln2 > 0. In addition to the fact, g(0) = 0 and lim f q g f = , so there exists a unique number fc > 0, where 0 < f* < fc < 1 such that g(fc) = 0. Finally, we may obtain the simplified Equation (9) by plugging Equation (7) into Equation (6).

    g ( f * ) = p l n p + q ln q + ln 2
    (9)

    The statement above says that the Kelly criterion of maximizing E[lnXN] is asymptotically optimal and the difference between the optimal Kelly strategy and others increases faster than the standard deviation of ln X N * ln X N securing P ( ln X N * ln X N ) > 0 1 (Brieman, 1961).

    If the Kelly criterion is applied to an investment portfolio consisting of a stock and a bond, yielding a return of rs with fs and rb with fb, respectively, Equation (6) may be modified as Equation (10). Here it is assumed that the returns of the two assets independently behave (Thorp, 1997).

    g ( f ) = p ln ( 1 + r s f s + r b f b ) + q ln ( 1 r s f s + r b f b )
    (10)

    And using the notations introduced above, Equation (10) can be equivalently expressed as(11)

    g ( f ) = p ln ( β s f s + β b f b ) + q l n ( 1 β s f s + β b f b ) ,    where    β s = 1 + r s   and   β b = 1 + r b
    (11)

    2.2.The Kelly Criterion and Deficiency in the Markowitz Portfolio Theory

    Hakansson (1971) and Thorp (1969) proved that the GM-V model was superior to the AM-V model in determining efficient portfolios if investment was made over a long period of time. Here are three interesting and unusual findings which Hakansson (1971) summarized in his paper: (1) the portfolio formed by the GM-V model does not lie even adjacent to the efficient frontier of the AM-V model, (2) the only feasible portfolios leading to ruin in the long run are those on the efficient frontier of the AM-V model, and (3) even the most undesirable portfolio on the GM-V model yields better return than most of the efficient portfolios on the AG-V model over the long run. However, Zhang (2010) proved that the portfolio on the AM-V efficient frontier stayed on the GM-V efficient model when the returns and variances of investment assets were small and borrowing was prohibited. If investors are faced with this case, they easily pick up the best portfolio with the the GM-V model, which has not been traditionally selected with the AM-V model.

    In the other aspect, Thorp (1971) demonstrated that the AM-V model was incongruent with the stochastic dominance concept. The AM-V model says that if E(P1) < E(P2) and Var(P1) > Var(P2), then a portfolio of P2 dominates a portfolio of P1 . Now, suppose that P1 and P2 are uniformly distributed on (Broadie, 1997; Fama and MacBeth, 1974) and (Markowitz, 1991; Thorp, 1969), respectively. Then, E(P1) = 4 < E(P2) = 20, and Var(P1) = 1/3 < Var(P2) = 25/3. From the AM-V model point of view, both portfolios do not dominate each other leading them to lie on the efficient frontier. However, investors obviously choose P2 over P1 on the contrary to our traditional knowledge about a stochastic dominance rule because there is no likelihood that the worst return of P2 falls below the best return of P1 , which follows the first-degree stochastic dominance rule.

    Next, presenting a short real example, we will show how to determine the best portfolio with the GM-V model, which can not be done with the AM-V model, and that even some of the efficient portfolios make investors completely get ruined in the long run.

    For this presentation, we selected two stocks: KAL (Korean Air Line) and S-Oil(one of the Korean petroleum companies) listed in KOSPI 200. Data on the returns and the corresponding probabilities regarding to two stocks are shown as in Table 1. For simplicity without the loss of generality, the returns of these two stocks are assumed to be independent each other even if there generally exits an inverse relationship between them.

    In the example, we will look at the case where the investment capital is completely allocated to the stocks. It means that f(KAL) and f(S-Oil) are constrained by(12)

    f ( K A L ) + f ( S O i l ) = 1
    (12)

    where f(KAL) and f(S-Oil) ≥ 0

    And the mean and variance of the return of the portfolio consisting of the two stocks are calculated using Equation (13) and (14), respectively.

    E ( P ) = 0.0077 f K A L + 0.1060 f S O i l
    (13)

    V a r ( P ) = 0.1760 f 2 K A L + 0.1203 f 2 S O i l
    (14)

    where E(KAL) = 0.0077, E(SOil) = 0.1060, Var(KAL) = 0.1760, and Var(SOil) = 0.1203.

    Figure 1 shows the set of the mean and standard deviation of portfolios constructed with the AM-V model, which are feasible subject to Equation 12. In the figure, Point A corresponds to a portfolio of (0, 1) which means that a full amount of an investment capital is allocated to S-Oil stock and nothing to KAL stock, while point I represents a portfolio of (1, 0). And point E indicates a portfolio of (0.4286, 0.5714) where 42.86% and 57.14% of the investment capital are allocated to KAL and S-Oil stocks, respectively. The curve ABCDE in Figure 1 forms an efficient frontier, i.e., all portfolios with 57.1430% or more of the investment capital allocated to the S-Oil stock. Refer to Table 2 for more information on each portfolio in Figure 1.

    Now, let us obtain the portfolios with the GM-V model maximizing the logarithmic expected return of Equation (15) subject to Equation (14).

    g ( f ) = E { ln [ X N X 0 ] 1 / N } = p ln ( β s f s + β b f b ) + q ln ( 1 β s f s + β b f b )
    (15)

    The optimal solution, f* , to the problem is (0.0305, 0.9695) which is represented by point B in Figure 1. The portfolio gives g(f*) = 0.0493 which is equivalent to e0.0476 =1.0505. Thus, the expected growth rate of the portfolio turns out to be approximately 5.05% which is higher than that of any AM-V efficient portfolio. As a result, investors can now select the best portfolio with the GM-V model which has a non-dominating relationship with other portfolios on the AM-V efficient frontier. Also, note that if the investors mistakenly choose one of portfolios among G, H, and I which lie on the inefficient frontier and whose returns are even positive in terms of the AM-V model, it is certain that they will be completely ruined in the long run.

    3.A CASE STUDY WITH KOSPI 200

    As mentioned in the beginning of the paper, the main purpose of the paper is to compare the results of the portfolios under the AM-V and GM-V models to see which one provides investors with a greater return and more useful information on their investing activities. The former is related to the well-known Markowitz portfolio model and the latter to the Kelly criterion which is new to most of us.

    Below, we will present two optimization models associated with the Kelly’s (GM-Vmodel) and Markowitz’s (AM-V model) portfolio problems which will be solved with LINGO 14, and analyze the outputs of the two models. To this end, we collected data on the stock prices of the companies listed in KOSPI 200 ranging from January of 1986 to August of 2015, and used it for the models. Table 3 contains a number of different types of data: the stock codes and names, a gain probability, a gain return, a loss return, and an expected return, while Table 4 involves a covariance matrix regarding to the stock prices.

    Since the Kelly criterion is concerned with the optimal allocation of the investment capital to maximize the expected geometric return on investment in the long run, maximizing the expected geometric return becomes the objective function of the Kelly portfolio model subject to the following two constraints. The first constraint in the model below says that an investment capital may be either fully or partially allocated to the stocks in the portfolios. Such a constraint is valid with the assumption that a borrowing and short sale activities are forbidden, which is corresponded with the second one. Taken these two constraints together, if there remains a portion of the investment capital which has not been allocated, it is assumed that the remaining portio is invested in risk-free government bonds yielding a return of r. Here, a betting ratio of f becomes a decision variable of the model.

    Maximize  i = 1 N { p i ln ( 1 + W i f i ) + ( 1 p i ) ln ( 1 L i f i ) }         + r ( 1 i = 1 N f i ) S u b j e c t t o i = 1 N f i 1 0 f i 1 ,  for  i = 1 , 2 , , N

    where

    • N : a number of the stocks in the portfolio

    • fi : a fraction of an investment capital allocated to stock i

    • r : a risk-free rate of return on one-year government bond

    • pi : a probability that the price of stock i increases

    • Wi : an expected return when the price of stock i increases

    • Li : an expected return when the price of stock i decreases.

    The optimization problem for the Markowitz portfolio is given below. The model may be formulated with an objective function of either maximizing an expected return or minimizing risk, i.e., a variance of thea portfolio, subject to the corresponding constraints. In this paper, we adopted maximizing the expected return of the portfolio instead of minimizing its risk. The first constraint in the model specifies the allowable maximum amount of risk, and the second one confirms that the total amount of an investment capital can not be greater than what investors hold on hand. If there is a remaining portion of the capital, it is assumed to be invested in a government bond yielding a risk-free rate of return of r as discussed in the GM-V model. Lastly, the third one says that no borrowing and short sales are allowed for each stock as in the GM-V model.

    Maximize  j = 1 N r j f j + r ( 1 j = 1 N f j ) S u b j e c t t o j = 1 N j = 1 N f i f j a i j = δ j = 1 N f j 1 0 f i 1 ,  for  i = 1 , 2 , , N

    where

    • N : a number of the stocks in the portfolio

    • fi : a fraction of an investment capital allocated to stock i

    • r : a risk-free rate of return on one-year government bond

    • σij : a covariance of the returns between stock i and j

    • δ : a maximum allowable amount of risk

    Table 5 and Table 6 present the results of the two portfolio optimization models. An interesting finding in the case study is that although we initially considered 200 stocks, two models selected only 3 and 5 stocks out of them, which is so different from our expectation. Comparing the expected returns of the two models, we can then see that the GM-V model gives investors the higher expected value of two ones by 0.0690% per week, while the former provides them with the greater risk by 0.0921% per week. However, the optimal solution of the GM-V model displays the less diversified number of the stocks than that of the AM-V model by 2 stocks in this particular illustrative example. This observation was not mathematically proved here, but a number of researchers such as Thorp (1997), Rotando and Thorp (1992), and Laureti et al. (2010) have shown that the observation is true.

    Correspoding to a degree of diversification, Estrada (2010) cites Grauer’s finding (1978) that neither of two models renders highly diversified portfolios. He also reports that the Kelly portfolios are less diversified than the Markowitz portfolios, and the former lends higher expected return and risk than the latter as stated above. According to Hunt (2005a, 2005b), the Kelly model presents the geometric return and risk as twice as the three benchmark portfolios: the equally-weighted portfolio, the minimum variance portfolio, and the portfolio with minimum risk for a targeted return of 15%. Many researchers including Brieman (1961), Estrada (2010), Hakansson (1971), and Thorp (1969) maintain that these are the characteristics of the two models. Therefore, we concluded with much caution that the results of our case study was also consistent with those of the previous researchers’ argument regarding to the Kelly models.

    4.CONCLUDING REMARKS

    If there exists 100% of probability that an expected rate of return greater than or equal to 0 is obtained, the best investment strategy is to put a full amount of investment capital into investment assets (e.g., stocks) to maximize profit. However, since an investment environment is so uncertain that the full betting investment strategy at once is not valid and efficient. Under such a real investment environment, the practical and effective investment strategy is to divide the investment capital into two parts: one part invested in risky assets and the other one in risk-free assets. The Kelly criterion model is just one of the best investment strategies which accomplish this goal. Poundstone (2006) and Thorp (2008) report such Shannon, Thorp, Ziemba, and Buffet as investors who had accomplished a high rate of return using the investment strategy

    We constructed the optimization portfolios with the stocks listed in the KOSPI 200 using the Kelly’s and a Markowitz’s model to empirically see the characteristics of two portfolio models. And it was found that investing in the stocks using the Kelly portfolio model generated much higher rate of return and risk on investment, and less diversified portfolio than a traditional investing strat- egies like the Markowitz portfolio model. As a further research, we need to develop more practical, sophisticated, and linearized version of the original nonlinear programming model shown in this paper to derive more useful investment information by analyzing a primal-dual relationship of a linear programming model.

    ACKNOWLEDGEMENT

    This study was supported by Korean Science and Engineering Foundation in 2012 (Project No.: NRF- 2012R1A1A4A01019650).

    Figure

    IEMS-16-392_F1.gif

    The efficient frontier with the AM-V model.

    Table

    The return and probability of KAL and S-Oil stocks

    The allocation ratio, mean, and standard deviation of the portfolios

    The basic data on the performance of a stock

    Covariance matrix

    The result of the optimal kelly portfolio

    The result of the optimal markowitz portfolio

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