Leonard C. MacLean

Dalhousie University, Halifax, Nova Scotia, Canada

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Publications (36)20.32 Total impact

  • 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.
    International Journal of Reliability Quality and Safety Engineering 10/2012; 19(04).
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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.
    SSRN Electronic Journal 01/2012;
  • The Journal of Portfolio Management 07/2011; 37(4):96-111. · 0.43 Impact Factor
  • 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.
    Quantitative Finance 04/2011; 10(7):681-687. · 0.75 Impact Factor
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    Ying Ma, Leonard Maclean, Kuan Xu, Yonggan Zhao
    Pacific Journal of Optimization 04/2011; 7(2). · 0.80 Impact Factor
  • 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.
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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.
  • 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
    12/2010: pages 3-20;
  • 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.
    11/2010: pages 277-296;
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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.
    SSRN Electronic Journal 03/2010;
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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.
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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 (1985) 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.
    Quantitative Finance 10/2009; · 0.75 Impact Factor
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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.
    The Journal of Risk Finance 01/2009;
  • 12/2007: pages 231-244;
  • 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.
    Journal of Banking & Finance 02/2007; · 1.29 Impact Factor
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    Leonard C Maclean, Yonggan Zhao, William T Ziemba
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    ABSTRACT: This paper presents a dynamic model of optimal currency returns with a hidden Markov regime switching process. We postulate a weak form of interest rate parity that the hedged risk premiums on currency investments are identical within each regime across all currencies. Both the in-sample and the out-of-sample data during January 2002 - March 2005 strongly support this hypothesis. Observing past asset returns, investors infer the prevailing regime of the economy and determine the most likely future direction to facilitate portfolio decisions. Using standard mean variance analysis, we find that an optimal portfolio resembles the Federal Exchange Rate Index which characterizes the strength of the U.S. dollar against world major currencies. The similarity provides a strong implication that our three-regime switching modelis appropriate for modeling the hedged returns in excess of the U.S. risk free interest rate. To investigate the impact of the equity market performance on changes of exchange rates, we include the S&P500 index return as an exogenous factor for parameter estimation.
    SSRN Electronic Journal 02/2006;
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    Leonard MacLean, Yonggan Zhao, William Ziemba
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    ABSTRACT: The risk inherent in the accumulation of investment capital depends on the true return distributions of the risky assets, the accuracy of estimated returns, and the investment strategy. This paper considers risk control with Value-at-Risk and Conditional Value-at-Risk, using control limits to determine times for portfolio rebalancing. Optimal strategies and control limits are determined for a geometric Brownian motion asset pricing model with random parameters. The approaches to risk control are applied to the fundamental problem of investment in stocks, bonds, and cash over time.
    Journal of Banking & Finance 02/2006; 30(2):317-339. · 1.29 Impact Factor
  • L. C. MacLean, W. T. Ziemba
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    ABSTRACT: Without Abstract
    01/2006: pages 448-456;
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    Leonard Maclean, William Ziemba, Yuming Li
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    ABSTRACT: This paper considers the problem of investment of capital in risky assets in a dynamic capital market in continuous time. The model controls risk, and in particular the risk associated with errors in the estimation of asset returns. The framework for investment risk is a geometric Brownian motion model for asset prices, with random rates of return. The information filtration process and the capital allocation decisions are considered separately. The filtration is based on a Bayesian model for asset prices, and an (empirical) Bayes estimator for current price dynamics is developed from the price history. Given the conditional price dynamics, investors allocate wealth to achieve their financial goals efficiently over time. The price updating and wealth reallocations occur when control limits on the wealth process are attained. A Bayesian fractional Kelly strategy is optimal at each rebalancing, assuming that the risky assets are jointly lognormal distributed. The strategy minimizes the expected time to the upper wealth limit while maintaining a high probability of reaching that goal before falling to a lower wealth limit. The fractional Kelly strategy is a blend of the log-optimal portfolio and cash and is equivalently represented by a negative power utility function, under the multivariate lognormal distribution assumption. By rebalancing when control limits are reached, the wealth goals approach provides greater control over downside risk and upside growth. The wealth goals approach with random rebalancing times is compared to the expected utility approach with fixed rebalancing times in an asset allocation problem involving stocks, bonds, and cash.
    Quantitative Finance 08/2005; 5(4):343-355. · 0.75 Impact Factor
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    ABSTRACT: This paper considers the estimation of parameters in a dynamic stochastic model for securities prices, where the expected rate of return is a random variable. An empirical Bayes estimator is developed from the model structure. The estimator is an improvement on other popular 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.
    SSRN Electronic Journal 05/2004;

Publication Stats

163 Citations
20.32 Total Impact Points


  • 1991–2012
    • Dalhousie University
      • Rowe School of Business
      Halifax, Nova Scotia, Canada
    • University of British Columbia - Vancouver
      • Sauder School of Business
      Vancouver, British Columbia, Canada