Handbook of the Fundamentals of Financial Decision Making

The following sections is included:

• Dedication
• Preface
• Contents
• About the Editors
• List of Contributors
• Acknowledgments

The theme of this handbook is financial decision making. The decisions are the amount of investment capital to allocate to various opportunities in a financial market. The opportunity set can be very complex, with sets of equities, bonds, commodities, derivatives, futures, and currencies having trading prices changing stochastically and dynamically over time. To consider decisions in a complex market, it is necessary to impose structure. In the abstract, the assets and the participants buying and selling them are parts of a system with underlying economic states. The system's dynamics and the factors defining the states in the system have been studied extensively in finance and economics. The dynamics of the market and the behavior of participants determine the trading prices of the various assets in the opportunity set…

The purpose of this paper is to examine rigorously the arbitrage model of capital asset pricing developed in Ross [13, 14]. The arbitrage model was proposed as an alternative to the mean variance capital asset pricing model, introduced by Sharpe, Lintner, and Treynor, that has become the major analytic tool for explaining phenomena observed in capital markets for risky assets. The principal relation that emerges from the mean variance model holds that for any asset, i, its (ex ante) expected return

The subsequent theorem is one of the pillars supporting the modern theory of Mathematical Finance…

The following sections are included:

• Introduction and History
• Risk neutral pricing for general models
• References

For valuing derivatives and other assets in securities and commodities markets, arbitrage pricing theory has been a major approach for decades. This paper derives fundamental arbitrage pricing results in finite dimensions in a simple unified framework using Tucker's theorem of the alternative. Frictionless results, that is perfect market results, plus imperfect market results such as those with dividends, periodic interest payments, transaction costs, different interest rates for lending and borrowing, shorting costs and constrained short selling are presented. While the results are mostly known and appear in various places, our contribution is to present them in a coherent and comprehensive fashion with very simple proofs. The analysis yields a simple procedure to prove new results and some arc presented for cases with imperfect market frictions.

Decision makers in financial markets are faced with a variety of opportunities and must decide how much capital to allocate to the various assets in the opportunity set at points in time. Investing in assets produces returns (gains/losses), resulting from the changes in trading prices. The returns on a unit of capital invested in assets are uncertain, that is, the return vector is a random variable. The basic information input to the decision on how much to invest in each asset is the distribution for the return vector. In Section A, some models for the dynamics of the stochastic return vector were considered. Assuming that the return distribution is known, the investment decisions are based on preferences for changes in wealth or accumulated wealth. To structure the decision process, a theory of preferences is required. The theory of preferences concerns the ability to represent a preference structure with a real-valued function. This has been achieved by mapping it to the mathematical index called utility. To put the preference relation for an individual into a theory of utility the following axioms were proposed by Von Neumann and Morgenstern (1944) and Savage (1954)…

The following sections are included:

• Introduction
• Definitions and notation
• Axioms and summary theorem
• Theorems
• Proof of Theorem 3
• Proof of Theorem 4
• Proof of Theorem 5
• Proof of Theorem 6
• REFERENCES

This paper presents a critique of expected utility theory as a descriptive model of decision making under risk, and develops an alternative model, called prospect theory. Choices among risky prospects exhibit several pervasive effects that are inconsistent with the basic tenets of utility theory. In particular, people underweight outcomes that are merely probable in comparison with outcomes that are obtained with certainty. This tendency, called the certainty effect, contributes to risk aversion in choices involving sure gains and to risk seeking in choices involving sure losses. In addition, people generally discard components that are shared by all prospects under consideration. This tendency, called the isolation effect, leads to inconsistent preferences when the same choice is presented in different forms. An alternative theory of choice is developed, in which value is assigned to gains and losses rather than to final assets and in which probabilities are replaced by decision weights. The value function is normally concave for gains, commonly convex for losses, and is generally steeper for losses than for gains. Decision weights are generally lower than the corresponding probabilities, except in the range of low probabilities. Overweighting of low probabilities may contribute to the attractiveness of both insurance and gambling.

Prospect theory is a paradigm challenging the expected utility paradigm. One of the fundamental components of prospect theory is the S-shaped value function. The value function is mainly justified by experimental investigation of the certainty equivalents of prospects confined either to the negative or to the positive domain, but not of mixed prospects, which characterize most actual investments. We conduct an experimental study with mixed prospects, using, for the first time, recently developed investment criteria called Prospect Stochastic Dominance (PSD) and Markowitz Stochastic Dominance (MSD). We reject the S-shaped value function, showing that at least 62%–76% of the subjects cannot be characterized by such preferences. We find support for the Markowitz utility function, which is a reversed S-shaped Function—exactly the opposite of the prospect theory value function. It is possible that the previous results supporting the S-shaped value function are distorted because the prospects had only positive or only negative outcomes, presenting hypothetical situations which individuals do not usually face, and which are certainly not common in financial markets.

Levy and Levy (Management Science 2002) present data that, according to their claims, violate prospect theory. They suggest that prospect theory's hypothesis of an S-shaped value function, concave for gains and convex for losses, is incorrect. However, all the data of Levy and Levy are perfectly consistent with the predictions of prospect theory, as can be verified by simply applying prospect theory formulas. The mistake of Levy and Levy is that they, incorrectly, thought that probability weighting could be ignored.

The experimental results of prospect theory (PT) reveal suggest that investors make decisions based on change of wealth rather than total wealth, that preference are S-shaped with a risk-seeking segment, and that probabilities are subjectively distorted. This article shows that while PT's findings are in sharp contradiction to the foundations of mean-variance (MV) analysis, counterintuitively, when diversification between assets is allowed, the MV and PT-efficient sets almost coincide. Thus one can employ the MV optimization algorithm to construct PTefficient portfolios…

In a classroom choice experiment with mixed gambles and moderate probabilities, we find severe violations of cumulative prospect theory (CPT) and of Markowitz stochastic dominance. Our results shed new light on the exchange between Levy and Levy (2002) and Wakker (2003) in this journal.

The following sections are included:

• Introduction
• Formulation
• Temporal von Neumann—Morgenstern Preference
• Analysis of the Two Period Model
• Analysis when Primitive Preference Is Temporal von Neumann—Morgenstern
• Approximating Induced Preference
• Acknowledgments
• References

This paper develops a class of recursive, but not necessarily expected utility, preferences over intertemporal consumption lotteries. An important feature of these general preferences is that they permit risk attitudes to be disentangled from the degree of intertemporal substitutability. Moreover, in an infinite horizon, representative agent context these preference specifications lead to a model of asset returns in which appropriate versions of both the atemporal CAPM and the intertemporal consumption-CAPM are nested as special cases. In our general model, systematic risk of an asset is determined by covariance with both the return to the market portfolio and consumption growth, while in each of the existing models only one of these factors plays a role. This result is achieved despite the homotheticity of preferences and the separability of consumption and portfolio decisions. Two other auxiliary analytical contributions which are of independent interest are the proofs of (i) the existence of recursive intertemporal utility functions, and (ii) the existence of optima to corresponding optimization problems. In proving (i), it is necessary to define a suitable domain for utility functions. This is achieved by extending the formulation of the space of temporal lotteries in Kreps and Porteus (1978) to an infinite horizon framework.

A final contribution is the integration into a temporal setting of a broad class of atemporal non-expected utility theories. For homogeneous members of the class due to Chew (1985) and Dekel (1986), the corresponding intertemporal asset pricing model is derived.

The following sections are included:

• INTRODUCTION
• SOME CALIBRATIONS BASED ON A THEOREM
• DISCUSSION AND CONCLUSION
• APPENDIX: THE THEOREM AND A COROLLARY
• REFERENCES

The following sections are included:

• Introduction
• The Expected Utility Model
• Systematic Violations of the Expected Utility Hypothesis
• Nonexpected Utility Functional Forms
• Generalized Expected Utility Analysis
• Applications to Insurance
• References

Many decisions are based on beliefs concerning the likelihood of uncertain events such as the outcome of an election, the guilt of a defendant, or the future value of the dollar. These beliefs are usually expressed in statements such as “I think that … ”, “chances are … ”, “it is unlikely that … ”, and so forth. Occasionally, beliefs concerning uncertain events are expressed in numerical form as odds or subjective probabilities. What determines such beliefs? How do people assess the probability of an uncertain event or the value of an uncertain quantity? This article shows that people rely on a limited number of heuristic principles which reduce the complex tasks of assessing probabilities and predicting values to simpler judgmental operations. In general, these heuristics are quite useful, but sometimes they lead to severe and systematic errors…

We discuss the cognitive and the psychophysical determinants of choice in risky and riskless contexts. The psychophysics of value induce risk aversion in the domain of gains and risk seeking in the domain of losses. The psychophysics of chance induce overweighting of sure things and of improbable events, relative to events of moderate probability. Decision problems can be described or framed in multiple ways that give rise to different preferences, contrary to the invariance criterion of rational choice. The process of mental accounting, in which people organize the outcomes of transactions, explains some anomalies of consumer behavior. In particular, the acceptability of an option can depend on whether a negative outcome is evaluated as a cost or as an uncompensated loss. The relation between decision values and experience values is discussed…

The ordering of individual preferences with utility or value functionals is dependent on the chosen utility. There are important almost canonical utilities. Bernoulli (1738) provided a rationale for the log utility, which together with the broader class of power utilities, has been useful in analyzing problems in decision making under uncertainty. Log utility is the basis of the golden rule of investing or Fortunes Formula, the Kelly (1956) strategy, which maximizes the asymptotic rate of capital growth. However, as indicated in Section B, there is some indeterminacy over individual utilities in practice and even over classes of utilities which reflect consistent preference attitudes…

The following sections are included:

• INTRODUCTION
• UNRESTRICTED UTILITY—THE GENERAL EFFICIENCY CRITERION
• EFFICIENCY IN THE FACE OF RISK AVERSION
• THE LIMITATIONS OF THE MEAN–VARIANCE EFFICIENCY CRITERION
• CONCLUSION
• REFERENCES

The theoretical and the empirical work in the theory of investment decision making under conditions of uncertainty (particularly in portfolio-selection theory) has been done so far under the very restricting assumption that all investors have a single target date for wealth accumulation, or investment horizon. They buy their portfolios on the same date with the object of maximizing their expected utility from terminal wealth. The papers of James Tobin and Jan Mossin who deal with the multi-period mean-variance criterion are the exception…

The components of the financial decision problem have been presented: assets, prices, utilities, and efficiency. However, there are important considerations in the decision models and the data inputs which affect the quality of investment decisions…

This paper concerns utility functions for money. A measure of risk aversion in the small, the risk premium or insurance premium for an arbitrary risk, and a natural concept of decreasing risk aversion are discussed and related to one another. Risks are also considered as a proportion of total assets.

This paper develops univariate and multivariate measures of risk aversion for correlated risks. We derive Rubinstein's measures of risk aversion from the risk premiums with correlated random initial wealth and risk. It is shown that these measures are not only consistent with those for uncorrelated or independent risks, but also have the corresponding local properties of the Arrow-Pratt measures of risk aversion. Thus Rubinstein's measures of risk aversion are the appropriate extension of the Arrow-Pratt measures of risk aversion in the univariate case. We also derive a risk aversion matrix from the risk premiums with correlated initial wealth and risk vectors. This matrix measure is the multivariate version of Rubinstein's measures and is also the generalization of Duncan's results for non-random initial wealth. The univariate and multivariate measures of risk aversion developed in this paper are applied to portfolio theory in Li and Ziemba [15].

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…

We colisider the .problem of portfolio selection for a risk averse investor wishing to allocate his resources among several investment opportunities in order to maximize the expected utility of final wealth. The calculation of the optimal investment proportions generally requires the solution of a stochastic program whose dimension is the number of risky investments. The computations are simplified dramatically when there is a risk free asset and the investment returns are jointly normally distributed. In this case Tobin has shown that the investment proportions in the risky assets are independent of the utility function and Lintner has shown that these proportions may be obtained from the solution of a fractional program. It is shown under mild hypotheses that the fractional program has a pseudo-concave objective and that the program always has a unique solution. The solution may be sought in several ways, perhaps most efficiently via Lemke's algorithm applied to a linear complementarity problem. The optimal investment proportions in all assets may be found by solving a stochastic program having one random variable and one decision variable via a search technique. Data on the major pooled Canadian equity pension funds were used to provide an empirical test of the suggested solution approach. Five common classes of utility functions were utilized with varying parameter values. For each class there are smooth curves that related the investment in the risk free asset to the parameters of the utility function. The investor is more risk averse when faced with quarterly data.

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 a version.

In a portfolio selection model with two risky investments having bivariate normally distributed returns, we show that Rubinstein's measures of risk aversion can yield the desirable characterizations of risk aversion and wealth effects on the optimal portfolios. These properties are analogous to those of the Arrow-Pratt measures of risk aversion in the portfolio selection model with one riskless and one risky investment. If investors' preferences are represented by multi-attributed utility functions and returns on different investments and other relevant factors have a joint normal distribution, we show that optimal portfolios can be characterized by a matrix measure of risk aversion.

This paper presents an efficient method for computing approximately optimal portfolios when the returns have symmetric stable distributions and there are many alternative investments. The procedure is valid, in particular, for independent investments and for multivariate investments of the classes introduced by Press and Samuelson. The algorithm is based on a two-stage decomposition of the problem and is analogous to the procedure developed by the author that is available for normally distributed investments utilizing Lintner's reformulation of Tobin's separation theorem…

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.

Economics can be distinguished from other social sciences by the belief that most (all?) behavior can be explained by assuming that rational agents with stable, well-defined preferences interact in markets that (eventually) clear. An empirical result qualifies as an anomaly if it is difficult to “rationalize” or if implausible assumptions are necessary to explain it within the paradigm. Suggestions for future topics should be sent to Richard Thaler, c/o Journal of Economic Perspectives, Graduate School of Business, University of Chicago, Chicago, IL 60637, or 〈thaler@gsb.uchicago.edu〉.

The following sections is included:

• Dedication
• Preface
• Contents
• About the Editors
• List of Contributors
• Acknowledgments

The concepts of utility and risk attitude such as decreasing risk aversion are general and qualitative. But risk is a basic to decision making models and requires a hard definition. The essential characteristics of risk are the chance of a potential loss and the size of the potential loss. If financial decisions are to be considered in the framework risk management, it is clear that a quantitative measure of risk is needed…

This paper describes the financial planning model TnnoALM we developed at Innovest for the Austrian pension fund of the electronics firm Siemens. The model uses a multi period 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 lnnoALM 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. lnnoALM 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 Australia, Siemens worldwide. and to evaluate possible pension fund regulation changes in Austria.

The notation in what follows is basically that of the paper of Artzner, Delbaen, Eber and Heath, but the axioms are less restrictive…

This paper provides an introduction to the theory of capital requirements defined by convex risk measures. The emphasis is on robust representations, law-invariant convex risk measures and their robustification in the face of model uncertainty, asymptotics for large portfolios, and on the connections of convex risk measures to actuarial premium principles and robust preferences.

This paper surveys the most recent advances in the context of decision making under uncertainty, with an emphasis on the modeling of risk-averse preferences using the apparatus of axiomatically defined risk functionals, such as coherent measures of risk and deviation measures, and their connection to utility theory, stochastic dominance, and other more established methods.

The components of financial decision making have been presented in previous sections: financial assets, trading prices, wealth preferences, decision sets, dominance, and efficiency. There is a rich literature on financial decision models, where variations on the components are proposed. Some of the models are considered in this final section. Most of the discussion is around power/log type utility functions and the associated capital growth. Within the framework of expected utility theory significant qualitative properties of decision behavior can be developed analytically for power utility, with log being the most risky power utility function…

Efficient market theory suggests that economic fundamentals of public firms are strongly associated with stock returns in the capital markets. However, there is no universal metric of fundamental strength of a firm to verify market efficiency. In this chapter, we develop a Data Envelopment Analysis-based relative strength evaluation technique using publicly-available annual accounting data for a given group of firms (or market). Under an iterative configuration of the data as inputs and outputs, we search for the maximum attainable correlation between finn strength and stock returns. Such a maximum correlation, termed firm Strength-Based Efficiency (SEE) index for the market, measures the degree to which firm fundamentals play a role in stock returns in a particular economy. Computing SI3E is a difficult discrete optimization problem that is solved via a hybrid simulated annealing procedure. SEE index values appear to be similar for the overall (large cap) markets in the U.S. and .Japan, and past (observed) firm strengths have no explanatory power on future stock returns. Expectations on future strengths, computed using Book-to-Market, Earnings-Per-Share, Leverage, Asset Turnover, etc., are evident in stock returns up to 2 years in advance, but market risk has only a negligible effect in our relative framework.

A strategy which has been of long standing interest in investing is the Kelly strategy, where the expected logarithm of wealth is maximized. There are many attractive properties when this strategy is used over a long planning horizon. Notably, the asymptotic growth rate of wealth is optimal and the time to reach asymptotically large wealth targets is minimized. However, the risk aversion of the Kelly strategy is essentially zero and the strategy is very risky in the short term. Investors can lose most of their wealth with a string of bad outcomes. In this paper the Kelly strategy and the associated fractional strategies are considered. The advantages and disadvantages are described in general and in particular with application to a variety of market scenarios. The conclusion discusses the use of Kelly type strategies by great investors.

We study a variety of optimal investment problems for objectives related to attaining goals by a fixed terminal time. We start by finding the policy that maximizes the probability of reaching a given wealth level by a given fixed terminal time, for the case where an investor can allocate his wealth at any time between n + l investment opportunities: n risky stocks, as well as a risk-free asset that has a positive return. This generalizes results recently obtained by Kulldorff and Heath for the case of a single investment opportunity. We then use this to solve related problems for cases where the investor has an external source of income, and where the investor is interested solely in beating the return of a given stochastic benchmark, as is sometimes the case in institutional money management One of the benchmarks we consider for this last problem is that of the return of the optimal growth policy, for which the resulting controlled process is a supermartingale. Nevertheless, we still find an optimal strategy. For the general case, we provide a thorough analysis of the optimal strategy, and obtain new insights into the behavior of the optimal policy. For one special case, namely that of a single stock with constant coefficients, the optimal policy is independent of the underlying drift. We explain this by exhibiting a correspondence between the probability maximizing results and the pricing and hedging of a particular derivative security, known as a digital or binary option. In fact, we show that for this case, the optimal policy to mOJ.imize the probability of reaching a given value of wealth by a predetermined time is equivalent to simply buying a European digital option with a particular strike price and payoff A similar result holds for the general case, but with the stock replaced by a particular (index) portfolio, namely the optimal growth or log-optimal portfolio.

We consider the portfolio problem in continuous-time where the objective of the investor or money manager is to exceed the performance of a given stochastic benchmark, as is often the case in institutional money management. The benchmark is driven by a stochastic process that need not be perfectly correlated with the investment opportunities, and so the market is in a sense incomplete. We first solve a variety of investment problems related to the achievement of goals: for example, we find the portfolio strategy that maximizes the probability that the return of the investor's portfolio beats the return of the benchmark by a given percentage without ever going below it by another predetermined percentage. We also consider objectives related to the minimization of the expected time until the investor beats the benchmark. We show that there are two cases to consider, depending upon the relative favorability of the benchmark to the investment opportunity the investor faces. The problem of maximizing the expected discounted reward of outperforming the benchmark, as well as minimizing the discounted penalty paid upon being outperformed by the benchmark is also discussed. We then solve a more standard expected utility maximization problem which allows new connections to be made between some specific utility functions and the nonstandard goal problems treated here.

We study stochastic dynamic investment games in continuous time between two investors (players) who have available two different, but possibly correlated, investment opportunities. There is a single payoff function which depends on both investors' wealth processes. One player chooses a dynamic portfolio strategy in order to maximize this expected payoff, while his opponent is simultaneously choosing a dynamic portfolio strategy so as to minimize the same quantity. This leads to a stochastic differential game with controlled drift and variance. For the most part, we consider games with payoffs that depend on the achievement of relative performance goals and/or shortfalls. We provide conditions under which a game with a general payoff function has an achievable value, and give an explicit representation for the value and resulting equilibrium portfolio strategies in that case. It is shown that non-perfect correlation is required to rule out trivial solutions. We then use this general result explicitly to solve a variety of specific games. For example, we solve a probability maximizing game, where each investor is trying to maximize the probability of beating the other's return by a given predetermined percentage. We also consider objectives related to the minimization or maximization of the expected time until one investor's return beats the other investor's return by a given percentage. Our results allow a new interpretation of the market price of risk in a Black-Scholes world. Games with discounting are also discussed, as are games of fixed duration related to utility maximization.

The Kelly criterion and fractional Kelly strategies hold an important place in investment management theory and practice. Both the Kelly criterion and fractional Kelly strategies, e.g. invest a fraction f of one's wealth in the Kelly portfolio and a proportion 1 − f in the risk-free asset, are optimal in the continuous time setting of the Merton [33] model. However, fractional Kelly strategies are no longer optimal when the basic assumptions of the Merton model, such as the lognormality of asset prices, are removed. In this chapter, we present an overview of some recent developments related to Kelly investment strategies in an incomplete market environment where asset prices are not lognormally distributed. We show how the definition of fractional Kelly strategies can be extended to guarantee optimality. The key idea is to get the definition of fractional Kelly strategies to coincide with the fund separation theorem related to the problem at hand. In these instances, fractional Kelly investment strategies appear as the natural solution for investors seeking to maximize the terminal power utility of their wealth.

The aim of this work is to extend the capital growth theory (Kelly, Latané, Breiman, Cover, Ziemba, Thorp, MacLean, Browne, Platen, Heath, and others) to asset market models with transaction costs. We define a natural generalization of the notion of a numeraire portfolio proposed by Long and show how such portfolios can be used for constructing growth-optimal investment strategies. The analysis is based on the classical von Neumann-Gale model of economic dynamics, a stochastic version of which we use as a framework for the modeling of financial markets with frictions.

We develop an approximate solution method for the optimal consumption and portfolio choice problem of an infinitely long-lived investor with Epstein–Zin utility who faces a set of asset returns described by a vector autoregression in returns and state variables. Empirical estimates in long-run annual and post-war quarterly U.S. data suggest that the predictability of stock returns greatly increases the optimal demand for stocks. The role of nominal bonds in long-term portfolios depends on the importance of real interest rate risk relative to other sources of risk. Long-term inflation-indexed bonds greatly increase the utility of conservative investors.

The following sections are included:

• Introduction
• Assumptions and Formulas for Mediocristan
• Geometric Growth, Standard Deviation and Sharpe Ratio
• Simulations
• Extremistan
• Geometric Growth, Standard Deviation and Sharpe Ratio
• Caveats and Conclusions
• Reference

The following sections is included:

• Author Index
• Subject Index