Leonard C. MacLean

Dalhousie University, Halifax, Nova Scotia, Canada

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Publications (40)24.57 Total impact

  • Leonard C. MacLean · William T. Ziemba

    No preview · Article · Jan 2016 · Wilmott
  • Leonard C. MacLean · W. T. Ziemba
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    ABSTRACT: To structure the decision process, a theory of preferences is required. The theory of preferences concerns the ability to represent structure with a real-valued function. This has been achieved by mapping it to the mathematical index call utility.
    No preview · Article · Nov 2015 · Wilmott
  • Leonard C. MacLean · Yonggan Zhao · William T. Ziemba
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    ABSTRACT: The optimal capital growth strategy or Kelly strategy has many desirable properties such as maximizing the asymptotic long-run growth of capital. However, it has considerable short-run risk since the utility is logarithmic, with essentially zero Arrow–Pratt risk aversion. It is common to control risk with a Value-at-Risk (VaR) constraint defined on the end of horizon wealth. A more effective approach is to impose a VaR constraint at each time on the wealth path. In this paper, we provide a method to obtain the maximum growth while staying above an ex-ante discrete time wealth path with high probability, where shortfalls below the path are penalized with a convex function of the shortfall. The effect of the path VaR condition and shortfall penalties is a lower growth rate than the Kelly strategy, but the downside risk is under control. The asset price dynamics are defined by a model with Markov transitions between several market regimes and geometric Brownian motion for prices within a regime. The stochastic investment model is reformulated as a deterministic programme which allows the calculation of the optimal constrained growth wagers at discrete points in time.
    No preview · Article · Aug 2015 · Quantitative Finance
  • Leonard C.maclean · Alexrichman
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    ABSTRACT: Complex systems are subject to failure with increased use and degradation. The risk process is the stochastic dynamic process of system failures and their severities. This paper considers aggregate risk measures for the risk process of complex systems in the context of stochastic ordering. The aggregation follows from the accumulation of losses from a series of failure events. The emphasis is on second-order risk measures which account for risk aversion as defined by concave utilities. A second-order measure termed the adjusted risk priority number (ARPN) is presented. The measure is constructed from well-known statistics: rate of failures, average severity of failures, and the Gini Index for severity of failures. The ARPN is contrasted with the traditional risk priority number (RPN) defined by the rate and average severity. The computation and use of the measures is illustrated with a spectrum of failure data from commercial aircraft in the USA.
    No preview · Article · Oct 2012 · International Journal of Reliability Quality and Safety Engineering
  • Leonard C. MacLean · Yonggan Zhao · William T. Ziemba
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    ABSTRACT: Covered interest rate parity assumes that there is no risk premium on the hedged returns on currencies. However, empirical evidence indicates that risk premiums are not identically zero, and this is referred to as the forward premium puzzle. We show that there exist market regimes, within which behavioral biases affect decisions, and a type of parity holds within regimes. The foreign exchange market switches between regimes where there is a premium. This paper presents various tests for the hypotheses of currency regimes and regime dependent risk premiums. Based on the existence of regimes, a diversified currency portfolio is created with a mean-variance criterion. Using the Federal Exchange Rate Index as a proxy for the currency benchmark and the U.S. T-Bill as the risk free asset, the similarity between the benchmarks and the implied equilibrium hedged and unhedged portfolios provides evidence for regimes and decision bias. Within each regime interest rate parity is appropriate for modeling currency returns.
    No preview · Article · May 2012 · Annals of Finance
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    Jun Yuan · Leonard MacLean · Kuan Xu · Yonggan Zhao
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    ABSTRACT: This paper explores how the returns of country exchange traded funds (ETFs) respond to global risk factors in different market regimes. We consider the ETFs for the U.S., Canada, U.K., Germany, France, Italy, Japan, and Australia from May 30, 2000 to March 31, 2011. To answer this question, we use the Bayesian information criterion to select a regime switching model (RS) with six global risk factors and identify three market regimes - bull, transitory and bear markets. The empirical results show that both the returns of country ETFs and their sensitivities to the risk factors are highly regime dependent. First, the U.S. size and value factors are significant in explaining most of selected ETFs across regimes. More specifically, small capitalization is associated with lower returns for all country ETFs (except for Canada) in at least one market regime. High book-to-market ratio generates higher returns for all ETFs in most market regimes. Second, the global stock market return has a positive impact on the returns of all country ETFs. Third, all ETFs returns are negatively correlated with market volatility in bull and bear market regimes. Fourth, a stronger U.S. dollar generates a higher return for the U.S. ETF and lower returns for the other seven country ETFs across market regimes. Finally, the returns of Australia, Canada and U.K. ETFs, which invest heavily in materials, are positively correlated with commodity prices while other country ETF returns are negatively associated with these prices across market regimes.
    Full-text · Article · Jan 2012 · SSRN Electronic Journal
  • Leonard C MacLean · Edward O Thorp · Yonggan Zhao · William T Ziemba
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    ABSTRACT: William Poundstone’s (2005) book, Fortune’s Formula, brought the Kelly capital growth criterion to the attention of investors. But how do full Kelly and fractional Kelly strategies that blend with cash actually preform in practice? To investigate this we revisit three simple investment situations and simulate the behavior of these strategies over medium term horizons using a large number of scenarios. These examples are from Bicksler and Thorp (1973) and Ziemba and Hausch (1986) and we consider many more scenarios and strategies. The results show: 1. the great superiority of full Kelly and close to full Kelly strategies over longer horizons with very large gains a large fraction of the time; 2. that the short term performance of Kelly and high fractional Kelly strategies is very risky; 3. that there is a consistent tradeoff of growth versus security as a function of the bet size determined by the various strategies; and 4. that no matter how favorable the investment opportunities are or how long the
    No preview · Article · Jul 2011 · The Journal of Portfolio Management
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    Leonard C. Maclean · Edward O. Thorp · William T. Ziemba
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    ABSTRACT: We summarize and discuss good and bad properties of the Kelly and fractionalKelly capital growth criteria. Additional properties are discussed as observations.
    Full-text · Article · Apr 2011 · Quantitative Finance
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    Ying Ma · Leonard Maclean · Kuan Xu · Yonggan Zhao
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    ABSTRACT: This paper develops a portfolio optimization model with a market neutral strategy under a Markov regime-switching framework. The selected investment instruments consist of the nine sector exchange traded funds (ETFs) that represent the U.S. stock market. The Bayesian information criterion is used to determine the optimal number of regimes. The investment objective is to dynamically maximize the portfolio alpha (excess return over the T-Bill) subject to neutralization of the portfolio sensitivities to the selected risk factors. The portfolio risk exposures are shown to change with various style and macro factors over time. The maximization problem in this context can be established as a regime-dependent linear programming problem. The optimal portfolio constructed as such is expected to outperform a naive benchmark strategy, which equally weights the ETFs. We evaluate the in-sample and out-of-sample performance of the regimedependent market neutral strategy against the equally weighted strategy. We find that the former generally outperforms the latter.
    Full-text · Article · Apr 2011 · Pacific Journal of Optimization
  • Leonard MacLean · Yonggan Zhao · William T. Ziemba
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    ABSTRACT: In capital growth under uncertainty, an investor must determine howmuch capital to invest in riskless and risky instruments at each point intime, with a focus on the trajectory of accumulated capital to a planninghorizon. Assuming that prices are not aected by individual investmentsbut rather aggregrate investments, individual decisions are made basedon the projected price process given the history of prices to date. An in-vestment strategy which has generated considerable interest is the growthoptimal or Kelly strategy, where the expected logarithm of wealth is maxi-mized period by period. In this paper the traditional capital growth modeland modications to control risk are developed. A mixture model basedon Markov transitions between normally distributed market regimes isused for the dynamics of asset prices. Decisions on investment in assetsare based on a constrained growth model, where the trajectory of wealthis required to exceed a specied path over time with high probability, andthe path violations are penalized using a convex loss function. This allowsthe determination of the optimal constrained growth wagers at discretepoints in time.
    No preview · Article · Mar 2011
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    Edward O. Thorp · Leonard MacLean · William T. Ziemba
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    ABSTRACT: This volume provides the definitive treatment of fortune's formula or the Kelly capital growth criterion as it is often called. The strategy is to maximize long run wealth of the investor by maximizing the period by period expected utility of wealth with a logarithmic utility function. Mathematical theorems show that only the log utility function maximizes asymptotic long run wealth and minimizes the expected time to arbitrary large goals. In general, the strategy is risky in the short term but as the number of bets increase, the Kelly bettor's wealth tends to be much larger than those with essentially different strategies. So most of the time, the Kelly bettor will have much more wealth than these other bettors but the Kelly strategy can lead to considerable losses a small percent of the time. There are ways to reduce this risk at the cost of lower expected final wealth using fractional Kelly strategies that blend the Kelly suggested wager with cash. The various classic reprinted papers and the new ones written specifically for this volume cover various aspects of the theory and practice of dynamic investing. Good and bad properties are discussed, as are fixed-mix and volatility induced growth strategies. The relationships with utility theory and the use of these ideas by great investors are featured.
    Full-text · Article · Feb 2011
  • William T. Ziemba · Leonard C. MacLean
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    ABSTRACT: This chapter describes the use of the Kelly capital growth model. This model, dubbed Fortune’s Formula by Thorp and used in the title by Poundstone (Fortune’s Formula: The Untold Story of the Scientific System That Beat the Casinos and Wall Street, 2005), has many attractive features such as the maximization of asymptotic long-run wealth; see Kelly (Bell System Technical Journal 35:917–926, 1956), Breiman (Proceedings of the 4th Berkely Symposium on Mathematical Statistics and Probability 1:63–68, 1961), Algoet and Cover (Annals of Probability 16(2):876–898, 1988) and Thorp (Handbook of Asset and Liability Management, 2006). Moreover, it minimizes the expected time to reach asymptotically large goals (Breiman, Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability 1:63–68, 1961) and the strategy is myopic (Hakansson, Journal of Business 44:324–334, 1971). While the strategy to maximize the expected logarithm of expected final wealth computed via a nonlinear program has a number of good short- and medium-term qualities (see MacLean, Thorp, and Ziemba, The Kelly Capital Growth Investment Critria, 2010b), it is actually very risky short term since its Arrow–Pratt risk aversion index is the reciprocal of wealth and that is essentially zero for non-bankrupt investors. The chapter traces the development and use of this strategy from the log utility formulation in 1738 by Bernoulli (Econometrica 22:23–36, 1954) to current use in financial markets, sports betting, and other applications. Fractional Kelly wagers that blend the E log maximizing strategy with cash tempers the risk and yield smoother wealth paths but with generally less final wealth. Great sensitivity to parameter estimates, especially the means, makes the strategy dangerous to those whose estimates are in error and leads them to poor betting and possible bankruptcy. Still, many investors with repeated investment periods and considerable wealth, such as Warren Buffett and George Soros, use strategies that approximate full Kelly which tends to place most of one’s wealth in a few assets and lead to many monthly losses but large final wealth most of the time. A simulation study is presented that shows the possibility of huge gains most of the time, possible losses no matter how good the investments appear to be, and possible extreme losses from overbetting when bad scenarios occur. The study and discussion shows that Samuelson’s objections to E log strategies are well understood. In practice, careful risk control or financial engineering is important to deal with short-term volatility and the design of good wealth paths with limited drawdowns. Properly implemented, the strategy used by many billionaires has much to commend it, especially with many repeated investments. KeywordsKelly investment criterion-Long-range investing-Logarithmic utility functions-Fractional Kelly strategies
    No preview · Chapter · Dec 2010
  • Leonard C. MacLean · Yonggan Zhao · William T. Ziemba
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    ABSTRACT: The accumulated wealth from investment in risky assets is a random variable. If investment strategies are to be ordered, so that one is preferred to another, then the ordering of random variables is required. In this paper the levels of stochastic dominance for random variables are used to define bi-criteria problems for determining an efficient investment strategy. The criteria are characterized as growth and security, respectively, and produce an ordering of strategies consistent with stochastic dominance. In the case where the dynamics of asset returns follow geometric Brownian motion in continuous time, the efficient strategies are shown to be proportional to the growth optimum or Kelly strategy. The analogous problem in discrete time requires solving a stochastic program. An example is provided which compares the continuous and discrete time solutions.
    No preview · Chapter · Nov 2010
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    Leonard MacLean · Yonggan Zhao · William T. Ziemba
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    ABSTRACT: We consider the presence of regimes in currency markets and their implications for interest rate parity. A weak form of interest rate parity is postulated and tested which assumes that the hedged risk premiums are identical within each regime across currencies. Both the in-sample (January 2002 - December 2004) and the out-of-sample (January 2005 - December 2007) daily data support weak interest rate parity. Furthermore, using the Federal Exchange Rate Index as a proxy of the currency market portfolio and T-Bills as the risk free asset, we find strong evidence that the weak interest rate parity hypothesis is consistent with standard portfolio equilibrium theory. The similarity between the benchmark and the implied equilibrium portfolio provides strong evidence that regime switching with weak interest rate parity is appropriate for modeling currency returns.
    Full-text · Article · Mar 2010 · SSRN Electronic Journal
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    Leonard C Maclean · Edward O Thorp · Yonggan Zhao · William T Ziemba
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    ABSTRACT: Using three simple investment situations, we simulate the behavior of the Kelly and fractional Kelly proportional betting strategies over medium term horizons using a large number of sce-narios. We extend the work of Bicksler and Thorp (1973) and Ziemba and Hausch (1986) to more scenarios and decision periods. The results show: (1) the great superiority of full Kelly and close to full Kelly strategies over longer horizons with very large gains a large fraction of the time; (2) that the short term performance of Kelly and high fractional Kelly strategies is very risky; (3) that there is a consistent tradeo of growth versus security as a function of the bet size determined by the various strategies; and (4) that no matter how favorable the investment opportunities are or how long the nite horizon is, a sequence of bad results can lead to poor nal wealth outcomes, with a loss of most of the investor's initial capital.
    Full-text · Article · Feb 2010
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    Leonard C. MacLean · Yonggan Zhao · William T. Ziemba
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    ABSTRACT: Standard delta hedging fails to exactly replicate a European call option in the presence of transaction costs. We study a pricing and hedging model similar to the delta hedging strategy with an endogenous volatility parameter for the calculation of delta over time. The endogenous volatility depends on both the transaction costs and the option strike prices. The optimal hedging volatility is calculated using the criterion of minimizing the weighted upside and downside replication errors. The endogenous volatility model with equal weights on the up and down replication errors yields an option premium close to the Leland [J. Finance, 198512. Leland , HE . 1985. Option pricing and replication with transaction costs. J. Finance, 40: 1283–1301. [CrossRef], [Web of Science ®]View all references, 40, 1283–1301] heuristic approach. The model with weights being the probabilities of the option's moneyness provides option prices closest to the actual prices. Option prices from the model are identical to the Black–Scholes option prices when transaction costs are zero. Data on S&P 500 index cash options from January to June 2008 illustrate the model.
    Full-text · Article · Oct 2009 · Quantitative Finance
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    Giorgio Consigli · Leonard C Maclean · Yonggan Zhao · William T Ziemba
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    ABSTRACT: 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 The trading prices for securities in financial markets can exhibit sudden shifts or reversals in direction. In this paper a methodology for asset price dynamics is presented where the diffusive component is combined with a risk process. The risk process accommodates deviations from an equilibrium process and reversions. The bond–stock yield differential is considered as a risk factor affecting the risk process. An approach using a "peaks over threshold" technique and conditional maximum likelihood is used to estimate parameters in the model. Numerical results for the period 1985–2004 in the US market validate the effectiveness of the model.
    Full-text · Article · Mar 2009 · The Journal of Risk Finance
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    Leonard C. MacLean · William T. Ziemba
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    ABSTRACT: In capital accumulation under uncertainty, a decision-maker must determine how much capital to invest in riskless and risky investment opportunities over time. The investment strategy yields a stream of capital, with investment decisions made so that the dynamic distribu- tion of wealth has desirable properties. The distribution of accumu- lated capital to a fixed point in time and the distribution of the first passage time to a fixed level of accumulated capital are variables con- trolled by the investment decisions. An investment strategy which has many attractive and some not attractive properties is the growth op- timal strategy, where the expected logarithm of wealth is maximized. This strategy is also referred to as the Kelly strategy It maximizes the rate of growth of accumulated capital.. With the Kelly strategy, the first passage time to arbitrary large wealth targets is minimized, and the probability of reaching those targets is maximized. However, the strategy is very aggressive since the Arrow-Pratt risk aversion index is essentially zero. Hence, the chances of losing a substantial portion of wealth are very high, particularly if the estimates of the returns distribution are in error. In the time domain, the chances are high
    Preview · Article · Dec 2008
  • Leonard MacLean · Yonggan Zhao · Giorgio Consigli · William Ziemba

    No preview · Chapter · Dec 2007
  • Leonard C. MacLean · Michael E. Foster · William T. Ziemba
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    ABSTRACT: This paper considers the estimation of the expected rate of return on a set of risky assets. The approach to estimation focuses on the covariance matrix for the returns. The structure in the covariance matrix determines shared information which is useful in estimating the mean return for each asset. An empirical Bayes estimator is developed using the covariance structure of the returns distribution. The estimator is an improvement on the maximum likelihood and Bayes–Stein estimators in terms of mean squared error. The effect of reduced estimation error on accumulated wealth is analyzed for the portfolio choice model with constant relative risk aversion utility.
    No preview · Article · Feb 2007 · Journal of Banking & Finance

Publication Stats

308 Citations
24.57 Total Impact Points


  • 1987-2015
    • Dalhousie University
      • • Rowe School of Business
      • • Department of Community Health & Epidemiology
      Halifax, Nova Scotia, Canada
  • 1992
    • Simon Fraser University
      Burnaby, British Columbia, Canada