There is considerable literature on the strengths and limitations of mean-variance analysis. The basic theory and extensions of MV analysis are discussed in Markowitz [1987] and Ziemba & Vickson [1975]. Bawa, Brown & Klein [1979] and Michaud [1989] review some of its problems…
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.
Click to increase image sizeClick to decrease image size Acknowledgements Special thanks go to Tom Cover, Michael Dempster and John Mulvey for helpful comments on an earlier draft of this paper. This is an expanded and updated discussion of a chapter in MacLean et al. (2010b MacLean, LC, Thorp, EO and Ziemba, WT editors. 2010b. The Kelly Capital Growth Investment Criterion: Theory and Practice, Singapore: World Scientific. [Google Scholar]).
This paper examines the effect of alternative utility functions and parameter values on the optimal composition of a risky investment portfolio. Normally distributed assets are the setting for the theoretical and empirical analyses. The results agree well with the available theory and imply utility functions and parameter values that are appropriate for investors with particular risk-bearing attitudes. The results give strong empirical support to the proposition that utility functions having different functional forms and parameter values but “similar” absolute risk aversion indices have “similar” optimal portfolios. These results suggest that over horizons up to one year one can safely substitute “convenient” surrogate utility functions for other utility functions, for reasons of tractability or otherwise. The results also provide guidance regarding the significance of the magnitude and change of particular numerical values of the risk aversion index. Moreover, theoretical (“exact”) results are obtained using Rubinstein's measure of global risk aversion.
This paper considers a portfolio problem with control on downside losses. Incorporating the worst-case portfolio outcome in the objective function, the optimal policy is equivalent to the hedging portfolio of a European option on a dynamic mutual fund that can be replicated by market primary assets. Applying the Black-Scholes formula, a closed-form solution is obtained when the utility function is HARA and asset prices follow a multivariate geometric Brownian motion. The analysis provides a useful method of converting an investment problem to an option pricing model.
World Scientific Series in FinanceCalendar Anomalies and Arbitrage, pp. 99-116 (2012) No AccessArbitrage Strategies for Cross-Track Betting on Major Horse RacesDonald B. Hausch and William T. ZiembaDonald B. HauschUniversity of Wisconsin—Madison, USA and William T. ZiembaUniversity of British Columbia, Canadahttps://doi.org/10.1142/9789814405461_0003Cited by:0 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail Abstract: The following sections are included: Introduction Efficiency of the Various Betting Markets Inefficiency of the Win Market and the Risk-free Hedging Model The Optimal Capital Growth Model Testing the One-Track Capital Growth Model Final Discussion References Without implicating them. we would like to thank Bruce Fauman and Fraser Rawlinson. Also. we greatly appreciate the data supplied by a number of U.S. racetracks, and we wish to thank Victor Lespinasse for suggesting the one-track model. FiguresReferencesRelatedDetails Calendar Anomalies and ArbitrageMetrics History PDF download
This paper is concerned with the determination of tight lower and upper bounds on the expectation of a convex function of a random variable. The classic bounds are those of Jensen and Edmundson-Madansky and were recently generalized by Ben-Tal and Hochman. This paper indicates how still sharper bounds may be generated based on the simple idea of sequentially applying the classic bounds to smaller and smaller subintervals of the range of the random variable. The bounds are applicable in the multivariate case if the random variables are independent. In the dependent case bounds based on the Edmundson-Madansky inequality are not available; however, bounds may be developed using the conditional form of Jensen's inequality. We give some examples to illustrate the geometrical interpretation and the calculations involved in the numerical determination of the new bounds. Special attention is given to the problem of maximizing a nonlinear program that has a stochastic objective function.
We investigate the stock market crashes in China, Iceland, and the US in the 2007-2009 period. The bond stock earnings yield difference model is used as a prediction tool. Historically, when the measure is too high, meaning that long bond interest rates are too high relative to the trailing earnings over price ratio, then there usually is a crash of 10% or more within four to twelve months. The model did in fact predict all three crashes. Iceland had a drop of fully 95%, China fell by two thirds and the US by 57%.
This paper concerns the problem of optimal dynamic choice in discrete time for an investor. In each period the investor is faced with one or more risky investments. The maximization of the expected logarithm of the period by period wealth, referred to as the Kelly criterion, is a very desirable investment procedure. It has many attractive properties, such as maximizing the asymptotic rate of growth of the investor's fortune. On the other hand, instead of focusing on maximal growth, one can develop strategies based on maximum security. For example, one can minimize the ruin probability subject to making a positive return or compute a confidence level of increasing the investor's initial fortune to a given final wealth goal. This paper is concerned with methods to combine these two approaches. We derive computational formulas for a variety of growth and security measures. Utilizing fractional Kelly strategies, we can develop a complete tradeoff of growth versus security. The theory is applicable to favorable investment situations such as blackjack, horseracing, lotto games, index and commodity futures and options trading. The results provide insight into how one should properly invest in these situations.
This paper discusses technical aspects of the Russell-Yasuda Kasai financial planning model. These include the models for the discrete distribution scenario generation processes for the uncertain parameters of the model, the mathematical approach used to develop the infinite-horizon end-effects part of the model, a comparison of algorithms used in the model's solution, and a comparison of the multistage stochastic linear programming model with the previous technology, static mean-variance analysis. Experience and benefits of the model in Yasuda-Kasai's financial planning process is also discussed.
This paper develops upper and lower bounds on two-stage stochastic linear programs using limited moment information. The case considered is when both the right-hand side as well as the objective coefficients of the second stage problem are random. Random variables are allowed to have arbitrary multivariate probability distributions with bounded support. First, upper and lower bounds are obtained using first and cross moments, from which we develop bounds using only first moments. The bounds are shown to solve the respective general moment problems.
In managing its assets and liabilities in light of uncertainties in cash flows, cost of funds and return on investments, a bank must determine its optimal trade-off between risk, return and liquidity. In this paper we develop a multiperiod stochastic linear programming model (ALM) that includes the essential institutional, legal, financial, and bank-related policy considerations, and their uncertainties, yet is computationally tractable for realistically sized problems. A version of the model was developed for the Vancouver City Savings Credit Union for a 5-year planning period. The results indicate that ALM is theoretically and operationally superior to a corresponding deterministic linear programming model, and that the effort required for the implementation of ALM, and its computational requirements, are comparable to those of the deterministic model. Moreover, the qualitative and quantitative characteristics of the solutions are sensitive to the model's stochastic elements, such as the asymmetry of cash flow distributions. We also compare ALM with the stochastic decision tree (SDT) model developed by S. P. Bradley and D. B. Crane. ALM is computationally more tractable on realistically sized problems than SDT, and simulation results indicate that ALM generates superior policies.
Many racetrack bettors have systems. Since the track is a market similar in many ways to the stock market one would expect that the basic strategies would be either fundamental or technical in nature. Fundamental strategies utilize past data available from racing forms, special sources, etc. to “handicap” races. The investor then wagers on one or more horses whose probability of winning exceeds that determined by the odds by an amount sufficient to overcome the track take. Technical systems require less information and only utilize current betting data. They attempt to find inefficiencies in the “market” and bet on such “overlays” when they have positive expected value. Previous studies and our data confirm that for win bets these inefficiencies, which exist for underbet favorites and overbet longshots, are not sufficiently great to result in positive profits. This paper describes a technical system for place and show betting for which it appears to be possible to make substantial positive profits and thus to demonstrate market inefficiency in a weak form sense. Estimated theoretical probabilities of all possible finishes are compared with the actual amounts bet to determine profitable betting situations. Since the amount bet influences the odds and theory suggests that to maximize long run growth a logarithmic utility function is appropriate the resulting model is a nonlinear program. Side calculations generally reduce the number of possible bets in any one race to three or less hence the actual optimization is quite simple. The system was tested on data from Santa Anita and Exhibition Park using exact and approximate solutions (that make the system operational at the track given the limited time available for placing bets) and found to produce substantial positive profits. A model is developed to demonstrate that the profits are not due to chance but rather to proper identification of market inefficiencies.
This paper describes the financial planning model InnoALM we developed at Innovest for the Austrian pension fund of the electronics firm Siemens. The model uses a multiperiod stochastic linear programming framework with a flexible number of time periods of varying length. Uncertainty is modeled using multiperiod discrete probability scenarios for random return and other model parameters. The correlations across asset classes, of bonds, stocks, cash, and other financial instruments, are state dependent using multiple correlation matrices that correspond to differing market conditions. This feature allows InnoALM to anticipate and react to severe as well as normal market conditions. Austrian pension law and policy considerations can be modeled as constraints in the optimization. The concave risk-averse preference function is to maximize the expected present value of terminal wealth at the specified horizon net of expected discounted convex (piecewise-linear) penalty costs for wealth and benchmark targets in each decision period. InnoALM has a user interface that provides visualization of key model outputs, the effect of input changes, growing pension benefits from increased deterministic wealth target violations, stochastic benchmark targets, security reserves, policy changes, etc. The solution process using the IBM OSL stochastic programming code is fast enough to generate virtually online decisions and results and allows for easy interaction of the user with the model to improve pension fund performance. The model has been used since 2000 for Siemens Austria, Siemens worldwide, and to evaluate possible pension fund regulation changes in Austria.
We study the land and stock markets in Japan circa 1990 and in 2013. While the Nikkei stock average in the late 1980s and its % crash in 1990 is generally recognized as a financial market bubble, a bigger bubble and crash was in the land market. The crash in the Nikkei which started on the first trading day of 1990 was predictable in April 1989 using the bond-stock earnings yield model which signalled a crash but not its exact moment. We show that it was possible to use the changepoint detection model based solely on price movements for profitable exits of long positions both circa 1990 and in 2013.
Multiasset portfolio selection models stated in terms of the expected utility criterion generally require the evaluation of multiple integrals. This reality has severely hindered attempts towards the development of computation methods to determine optimal portfolio allocations when there are a large number of assets. Aside from special cases, expected utility is not convergent into a simple closed form; the complexity from the point of view of computation is then perhaps most easily appreciated if one realizes that every iteration in a nonlinear program demands the estimation of several integrals (see Ziemba [23] for details). Such calculations are extremely costly when the number of assets is large. It is, consequently, of interest to approximate the expected utility function by a function which is easier to optimize over the set of feasible portfolios.
In many nonlinear programming applications the objective function has an inherent uncertainty that depends upon a set of random variables that have a known distribution. If one wishes to optimize the expectation of the objective, as suggested by the expected utility theorem, then as is shown here one can often solve such problems by modifying standard nonlinear programming algorithms. To illustrate what is involved, the details and justification for the application of the interior parametric sequential unconstrained maximization technique and the generalized programming method for the solution of such problems are given. Some related problems with stochastic constraints for which the solution method applies are mentioned and an example of a portfolio selection problem is given.
