DOCSLIB.ORG

Camp-FM 1..14

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Camp-FM 1..14 Contents List of Figures xiii List of Tables xiv 1. Introduction 1 2. Myopic Portfolio Choice 17 2.1. Short-Term Portfolio Choice 18 2.2. Myopic Long-Term Portfolio Choice 31 2.3. Conclusion 46 3. Who Should Buy Long-Term Bonds? 48 3.1. Long-Term Portfolio Choice in a Model with Constant Variances and RiskPremia 50 3.2. A Model of the Term Structure of Interest Rates 62 3.3. Conclusion: Bonds, James, Bonds 86 4. Is the Stock Market Safer for Long-Term Investors? 88 4.1. Long-Term Portfolio Choice in a VAR Model 90 4.2. Stockand Bond MarketRiskin Historical US Data 101 4.3. Conclusion 117 5. Strategic Asset Allocation in Continuous Time 120 5.1. The Dynamic Programming Approach 121 5.2. The Martingale Approach 133 5.3. Recursive Utility in Continuous Time 143 5.4. Should Long-Term Investors Hedge Volatility Risk? 147 5.5. Parameter Uncertainty and Portfolio Choice 154 5.6. Conclusion 160 xii Contents 6. Human Wealth and Financial Wealth 162 6.1. Single-Period Models with Labor Income 167 6.2. Labor Income, Precautionary Savings, and Long-Horizon Portfolio Choice 182 6.3. Conclusion 193 7. Investing over the Life Cycle 195 7.1. What Do We Know about Household Asset Allocation? 197 7.2. A Life-Cycle Model of Portfolio Choice 202 7.3. Conclusion 219 References 226 Author Index 241 Subject Index 249 1 Introduction One of the most important decisions many people face is the choice of a portfolio of assets for retirement savings. These assets may be held as a supplement to de®ned-bene®t public or private pension plans; or they may be accumulated in a de®ned- contribution pension plan, as the major source of retirement income. In either case, a dizzying array of assets is available. Consider for example the increasing set of choices offered by TIAA±CREF, the principal pension organization for university employees in the United States. Until 1988, the two available choices were TIAA, a traditional nominal annuity, and CREF, an actively managed equity fund. Funds could readily be moved from CREF to TIAA, but the reverse transfer was dif®cult and could be accomplished only gradually. In 1988 it became possible to move funds between two CREF accounts, a money market fund and an equity fund. Since then other choices have been added: a bond fund and a socially responsible stock fund in 1990, a global equity fund in 1992, equity indexand growth funds in 1994, a real estate fund in 1995, and an in¯ation-indexed bond fund in 1997. Retirement savings can easily be moved among these funds, each of which represents a broad class of assets with a different pro®le of returns. Institutional investors also face complexdecisions. Some institutions invest on behalf of their clients, but others, such as foundations and university endowments, are similar to indivi- duals in that they seek to ®nance a long-term stream of discre- tionary spending. The investment options for these institutions have also expanded enormously since the days when a portfolio of government bonds was the norm. Mean±variance analysis What does ®nancial economics have to say about these invest- ment decisions? Modern ®nance theory is often thought to have started with the mean±variance analysis of Markowitz (1952); this 2 Introduction Expected return Aggressive investor Stocks Moderate Best mix of investor stocks and bonds Conservative Bonds investor Cash Risk Figure 1.1. Mean±standard deviation diagram. makes portfolio choice theory the original subject of modern ®n- ance. Markowitz showed how investors should pick assets if they care only about the mean and varianceÐor equivalently the mean and standard deviationÐof portfolio returns over a single period. The results of his analysis are shown in the classic mean± standard deviation diagram, Figure 1.1. (A much more careful mathematical explanation can be found in the next chapter.) For simplicity the ®gure considers three assets: stocks, bonds, and cash (not literally currency, but a short-term money market fund). The vertical axis shows expected return, and the horizontal axis shows risk as measured by standard deviation. Stocks are shown as offering a high mean return and a high standard deviation, bonds a lower mean and lower standard deviation. Cash has a lower mean return again, but is riskless over one period, so it is plotted on the vertical zero-risk axis. (In the presence of in¯ation risk, nominal money market investments are not literally riskless in real terms, but this short-term in¯ation risk is small enough that it is conventional to ignore it. We follow this convention here and return to the issue in the next chapter.) The curved line in Figure 1.1 shows the set of means and standard deviations that can be achieved by combining stocks Introduction 3 and bonds in a risky portfolio. When cash is added to a portfolio of risky assets, the set of means and standard deviations that can be achieved is a straight line on the diagram connecting cash to the risky portfolio. An investor who cares only about the mean and standard deviation of his portfolio will choose a point on the straight line illustrated in the ®gure, that is tangent to the curved line. This straight line, the mean±variance ef®cient frontier, offers the highest mean return for any given standard deviation. The point where the straight line touches the curved line is a `tangency portfolio' of risky assets, marked in the ®gure as `Best mixof stocks and bonds'. The striking conclusion of this analysis is that all investors who care only about mean and standard deviation will hold the same portfolio of risky assets, the unique best mixof stocks and bonds. Conservative investors will combine this portfolio with cash to achieve a point on the mean±variance ef®cient frontier that is low down and to the left; moderate investors will reduce their cash holdings, moving up and to the right; aggressive investors may even borrow to leverage their holdings of the tangency portfolio, reaching a point on the straight line that is even riskier than the tangency portfolio. But none of these investors should alter the relative proportions of risky assets in the tangency portfolio. This result is the mutual fund theorem of Tobin (1958). Financial planning advice Financial planners have traditionally resisted the simple invest- ment advice embodied in Figure 1.1. This resistance may to some extent be self-serving; as Peter Bernstein points out in his 1992 book Capital Ideas, many ®nancial planners and advisors justify their fees by emphasizing the need for each investor to build a portfolio re¯ecting his or her unique personal situation. Bernstein calls this the `interior decorator fallacy', the view that portfolios should re¯ect personal characteristics in the same way that interior decor re¯ects personal taste. There are however many legitimate reasons why different portfolios of risky assets might be appropriate for different investors. The complexity of the tax code creates many differ- ences across investors, not only in their taxbrackets but also in their opportunities for sheltering income from taxation. Beyond this, investors differ in their investment horizons: some may have 4 Introduction relatively short-term objectives, others may be saving to make college tuition payments in the medium term, yet others may be saving for retirement or to ensure the well-being of their heirs. Investors also differ in the characteristics of their labor income: young investors may expect many years of income, which may be relatively safe for some and risky for others, while older investors may need to ®nance the bulk of their consumption from accumulated ®nancial wealth. Investors often have illiquid assets such as family businesses, restricted stock options granted by their employer, or real estate.1 We shall argue in this book that the traditional academic analysis of portfolio choice needs to be modi®ed to handle long investment horizons, labor income, and illiquid assets. The modi®ed theory explains several of the patterns that we see in conventional ®nancial planning advice. One strong pattern is that ®nancial planners typically encou- rage young investors, with a long investment horizon, to take more risk than older investors. The single-period mean±variance analysis illustrated in Figure 1.1 assumes a short investment horizon. In this book we shall explore the conditions under which a long investment horizon does indeed justify greater risk-taking. A second pattern in ®nancial planning advice is that con- servative investors are typically encouraged to hold more bonds, relative to stocks, than aggressive investors, contrary to the constant bond±stock ratio illustrated in Figure 1.1. Canner, Mankiw, and Weil (1997) call this the asset allocation puzzle. Table 1.1, which reproduces Table 1 from Canner et al.'s article, illustrates the puzzle. The table summarizes model portfolios recommended by four different investment advisors in the early 1990s: Fidelity, Merrill Lynch, the ®nancial journalist Jane Bryant Quinn, and the New York Times. While the portfolios differ in their details, in every case the recommended ratio of bonds to stocks is 1 Perhaps some of these differences underlie the PaineWebber advertisement that ran in the New Yorker and other magazines in 1998: `If our clients were all the same, their portfolios would be too. They say the research has been sifted. The numbers have been crunched. The analysts have spoken: Behold! The ideal portfolio. We say building a portfolio is not ``one size ®ts all''. It begins with knowing youÐhow you feel about money, how much risk you can tolerate, your hopes for your family, and for your future.
Load more

Recommended publications
  • Portfolio Selection Under Exponential and Quadratic Utility
    Portfolio Selection Under Exponential and Quadratic Utility Steven T. Buccola The production or marketing portfolio that is optimal under the assumption of quadratic utility may or may not be optimal under the assumption of exponential utility. In certain cases, the necessary and sufficient condition for an identical solution is that absolute risk aversion coefficients associated with the two utility functions be the same. In other cases, equality of risk aversion coefficients is a sufficient condition only. A comparison is made between use of exponential and quadratic utility in the analysis of a California farmer's marketing problem. A large number of utility functional forms difference. Under other conditions, the have been proposed for use in selecting opti- choice has a significant impact on optimal mal portfolios of risky prospects [Tsaing]. For behavior. purposes of both theoretical and applied eco- Suppose an analyst is studying a continu- nomic research, attention has primarily been ously divisible set of alternative production directed to the quadratic and exponential or marketing options offering approximately forms. After enjoying widespread use in the normally-distributed returns, and that he 1950s and 1960s, the quadratic form came wishes to select the portfolio of these options under criticism by Pratt and others and lost with highest expected utility. The analyst is much of its reputability. In contrast the expo- considering using an exponential utility func- nential form has become increasingly popu- tion U = -exp(-Xx), X > 0, or a quadratic lar, due in large measure to its mathematical utility function U = x -vx,v, > 0, x < /2v, tractability [Attanasi and Karlinger; O'Con- where U is utility and x is wealth.
    [Show full text]
  • Utilitarianism, Expected Utility Theory, Catastrophic Risks JEL-Classification: Q54
    EXPECTED UTILITY THEORY AND THE TYRANNY OF CATASTROPHIC RISKS WOLFGANG BUCHHOLZ∗ AND MICHAEL SCHYMURAy FIRST VERSION: SEPTEMBER 01, 2010 THIS VERSION: DECEMBER 06, 2010 ABSTRACT - Expected Utility theory is not only applied to individual choices but also to eth- ical decisions, e.g. in cost-benefit analysis of climate change policy measures that affect future generations. In this context the crucial question arises whether EU theory is able to deal with "catastrophic risks", i.e. risks of high, but very unlikely losses, in an ethically appealing way. In this paper we show that this is not the case. Rather, if in the framework of EU theory a plausi- ble level of risk aversion is assumed, a "tyranny of catastrophic risk" (TCR) emerges, i.e. project evaluation may be dominated by the catastrophic event even if its probability is negligibly small. With low degrees of risk aversion, however the catastrophic risk eventually has no impact at all when its probability goes to zero which is ethically not acceptable as well. Keywords: Utilitarianism, Expected Utility Theory, Catastrophic Risks JEL-Classification: Q54 ∗University of Regensburg, Universitätsstraße 31, D-93053 Regensburg, Germany; Email: [email protected] yCentre for European Economic Research (ZEW), L7,1, D-68161 Mannheim, Germany; Email: [email protected]. The authors express their thank for helpful comments from John Roemer (Yale University), Martin Weitzman (Harvard University) and Kenneth Arrow (Stanford University). Furthermore this paper has benefited from the discussions at the 18th EAERE Annual Confer- ence in Rome and the 16th World Congress of the International Economic Association in Beijing.
    [Show full text]
  • Utility with Decreasing Risk Aversion
    144 UTILITY WITH DECREASING RISK AVERSION GARY G. VENTER Abstract Utility theory is discussed as a basis for premium calculation. Desirable features of utility functions are enumerated, including decreasing absolute risk aversion. Examples are given of functions meeting this requirement. Calculating premiums for simplified risk situations is advanced as a step towards selecting a specific utility function. An example of a more typical portfolio pricing problem is included. “The large rattling dice exhilarate me as torrents borne on a precipice flowing in a desert. To the winning player they are tipped with honey, slaying hirri in return by taking away the gambler’s all. Giving serious attention to my advice, play not with dice: pursue agriculture: delight in wealth so acquired.” KAVASHA Rig Veda X.3:5 Avoidance of risk situations has been regarded as prudent throughout history, but individuals with a preference for risk are also known. For many decision makers, the value of different potential levels of wealth is apparently not strictly proportional to the wealth level itself. A mathematical device to treat this is the utility function, which assigns a value to each wealth level. Thus, a 50-50 chance at double or nothing on your wealth level may or may not be felt equivalent to maintaining your present level; however, a 50-50 chance at nothing or the value of wealth that would double your utility (if such a value existed) would be equivalent to maintaining the present level, assuming that the utility of zero wealth is zero. This is more or less by definition, as the utility function is set up to make such comparisons possible.
    [Show full text]
  • The Case Against Power Utility and a Suggested Alternative: Resurrecting Exponential Utility
    Munich Personal RePEc Archive The Case Against Power Utility and a Suggested Alternative: Resurrecting Exponential Utility Alpanda, Sami and Woglom, Geoffrey Amherst College, Amherst College July 2007 Online at https://mpra.ub.uni-muenchen.de/5897/ MPRA Paper No. 5897, posted 23 Nov 2007 06:09 UTC The Case against Power Utility and a Suggested Alternative: Resurrecting Exponential Utility Sami Alpanda Geoffrey Woglom Amherst College Amherst College Abstract Utility modeled as a power function is commonly used in the literature despite the fact that it is unbounded and generates asset pricing puzzles. The unboundedness property leads to St. Petersburg paradox issues and indifference to compound gambles, but these problems have largely been ignored. The asset pricing puzzles have been solved by introducing habit formation to the usual power utility. Given these issues, we believe it is time re-examine exponential utility. Exponential utility was abandoned largely because it implies increasing relative risk aversion in a cross-section of individuals and non- stationarity of the aggregate consumption to wealth ratio, contradicting macroeconomic data. We propose an alternative preference specification with exponential utility and relative habit formation. We show that this utility function is bounded, consistent with asset pricing facts, generates near-constant relative risk aversion in a cross-section of individuals and a stationary ratio of aggregate consumption to wealth. Keywords: unbounded utility, asset pricing puzzles, habit formation JEL Classification: D810, G110 1 1. Introduction Exponential utility functions have been mostly abandoned from economic theory despite their analytical convenience. On the other hand, power utility, which is also highly tractable, has become the workhorse of modern macroeconomics and asset pricing.
    [Show full text]
  • Pareto Utility
    Theory Dec. (2013) 75:43–57 DOI 10.1007/s11238-012-9293-8 Pareto utility Masako Ikefuji · Roger J. A. Laeven · Jan R. Magnus · Chris Muris Received: 2 September 2011 / Accepted: 3 January 2012 / Published online: 26 January 2012 © The Author(s) 2012. This article is published with open access at Springerlink.com Abstract In searching for an appropriate utility function in the expected utility framework, we formulate four properties that we want the utility function to satisfy. We conduct a search for such a function, and we identify Pareto utility as a func- tion satisfying all four desired properties. Pareto utility is a flexible yet simple and parsimonious two-parameter family. It exhibits decreasing absolute risk aversion and increasing but bounded relative risk aversion. It is applicable irrespective of the prob- ability distribution relevant to the prospect to be evaluated. Pareto utility is therefore particularly suited for catastrophic risk analysis. A new and related class of generalized exponential (gexpo) utility functions is also studied. This class is particularly relevant in situations where absolute risk tolerance is thought to be concave rather than linear. Keywords Parametric utility · Hyperbolic absolute risk aversion (HARA) · Exponential utility · Power utility M. Ikefuji Institute of Social and Economic Research, Osaka University, Osaka, Japan M. Ikefuji Department of Environmental and Business Economics, University of Southern Denmark, Esbjerg, Denmark e-mail: [email protected] R. J. A. Laeven (B) · J. R. Magnus Department of Econometrics & Operations Research, Tilburg University, Tilburg, The Netherlands e-mail: [email protected] J. R. Magnus e-mail: [email protected] C.
    [Show full text]
  • Evolution of the Arrow–Pratt Measure of Risk-Tolerance for Predictable Forward Utility Processes
    Finance Stoch https://doi.org/10.1007/s00780-020-00444-1 Evolution of the Arrow–Pratt measure of risk-tolerance for predictable forward utility processes Moris S. Strub1 · Xun Yu Zhou2 Received: 18 September 2019 / Accepted: 26 August 2020 © The Author(s), under exclusive licence to Springer-Verlag GmbH, DE part of Springer Nature 2020 Abstract We study the evolution of the Arrow–Pratt measure of risk-tolerance in the framework of discrete-time predictable forward utility processes in a complete semi- martingale financial market. An agent starts with an initial utility function, which is then sequentially updated forward at discrete times under the guidance of a martin- gale optimality principle. We mostly consider a one-period framework and first show that solving the associated inverse investment problem is equivalent to solving some generalised integral equations for the inverse marginal function or for the conjugate function, both associated with the forward utility. We then completely characterise the class of forward utility pairs that can have a time-invariant measure of risk-tolerance and thus a preservation of preferences in time. Next, we show that in general, pref- erences vary over time and that whether the agent becomes more or less tolerant to risk is related to the curvature of the measure of risk-tolerance of the forward utility pair. Finally, to illustrate the obtained general results, we present an example in a bi- nomial market model where the initial utility function belongs to the SAHARA class, and we find that this class is analytically tractable and stable in the sense that all the subsequent utility functions belong to the same class as the initial one.
    [Show full text]
  • Estimating Exponential Utility Functions
    ,~ j,;' I) ':;l) . G rio"". 0 • 0 (? _.0 o I') Q () l% a 9(;' o Q estimatiqg "ExPC?ttenti~1 ·:U····:t·····~·I·····I:·I·+u·· ;F;";;u::·:n6',.t5"'·;~:n·'.' ·s.. ·, .' ..., .'.: :','. ..•. ".. '..'. ;. ~':,,;'V~ ..' .'. o c:::::::> ' ,;", o o (b~s. ) EconOm:ics, .Statistics'·Oftd (OOfI,ati,tS /} o i_JS~~n;'~~t'~C:'T" " o .j o i) " (> C. "French " State Un I vers" I ty, and -­ ~~ ~ from Agric~ltur;al Economic Research, Janucrry 1978" \/01. 30, No. 1 ~~'~....' .. ' .. exponential.utility fun~tion for money has 10ng att:racted attentic;m fromAtheorists~~ because it exhibits\~oni"creasing absolute, risk aversion. j\lso, under certain con- ~ ditions, it generates an expected utility functi()n that is maximizable in a quadratic 1 program. However, "this "'functiona,l form presents 'estimation problems. Logarithmic • transformation of an exponential uti 1i,t,y, function does not conform to the Von Neumann­ Horgenste~ axioms. Hence', it cannot be used as ,.a basis for best fit in statistical analysis. A criterion is described that can be used to select a best-flt exponential uti 1ity.fun.ction, and its appl.ica,ti~!!c"",L~I.J~~~er utJ1ity an~,,~J~!s. is (~en~~~.~~r.~~ed ,~-».,.....'~\ ~ ....,.,." .,-~~ '1\, ~.... jt-~ .. -~..,q,·""·"~.""""11!"~""""'"""'\.'<'..".tft4._ ... 1' ,,''-,. < ~ i\ ,c,( ~--~""..-~,."".... .... ,~. '" ~. "~:"''''.jq''K>''''~~''l;:' ", " & ' (J 11' 1 i ,~ r.~ .... ' t ' .'~ >':J ~rr 1 C >J '¥ ~r ~1 f' if! 1 t::' ~ t,n t; ~,l IG " Canputerized simulation o Computers c Economic analysis Est imates 1\, Exponential functions " ~ogarlthm functions Risk ~Statistical analysis Utility routines 1711. IdenEifier./Open-Eaded Titnu (;:::; Absolute risk aversion Compute r app,l i cat i ens' Logari thmi c traosformatian Honey (/ Von Neumann-Morgenstern axioms )) ~ 12-A, 12-8 PB .
    [Show full text]
  • Cardinal Sins: Utility Specification and the Measurement of Risk Aversion
    19 Eisenhauer, International Journal of Applied Economics, 13(2), September 2016, 19-29 Cardinal Sins: Utility Specification and the Measurement of Risk Aversion Joseph G. Eisenhauer* University of Detroit Mercy ___________________________________________________________________________ Abstract: In contrast to the ideal of allowing empirical data to reveal individual preferences, economists have frequently resorted to constructing preference functions with convenient parameters and then using data merely to estimate the magnitude of the parameters. In particular, it has become common practice for researchers seeking to measure risk aversion to select an arbitrary utility specification and assume that the same functional form applies to all individuals in a given sample. This note illustrates the prevalence of this practice and how unreliable such an approach may be. Using a recently developed data set from Thailand, empirical results for the Pratt-Arrow coefficients of risk aversion are compared across isoelastic and exponential utility functions. The latter are estimated to be, on average, more than five times larger than the former, suggesting the possibility of enormous distortions in empirical studies that assume specific cardinal utility functions. Keywords: risk aversion, utility function, isoelastic, exponential, representative agent JEL Classification: D810 ___________________________________________________________________________ 1. Introduction Although utility is canonically described as an ordinal ranking of preferences over goods and services, economists have frequently adopted specific functional forms that yield cardinal measures (Barnett, 2003). As early as the 18th century, Bernoulli (1738) used a logarithmic utility function to illustrate the concept of the diminishing marginal utility of wealth in his explanation of risk aversion. This tendency has become increasingly prevalent since Pratt (1964) and Arrow (1965) introduced quantitative measures of risk aversion.
    [Show full text]
  • Risk Aversion, Wealth, and Background Risk
    RISK AVERSION, WEALTH, AND BACKGROUND RISK Luigi Guiso Monica Paiella European University Institute, University of Naples Parthenope Ente Luigi Einaudi Abstract We use household survey data to construct a direct measure of absolute risk aversion based on the maximum price a consumer is willing to pay for a risky security. We relate this measure to consumer’s endowments and attributes and to measures of background risk and liquidity constraints. We find that risk aversion is a decreasing function of the endowment—thus rejecting CARA preferences. We estimate the elasticity of risk aversion to consumption at about 0.7, below the unitary value predicted by CRRA utility. We also find that households’ attributes are of little help in predicting their degree of risk aversion, which is characterized by massive unexplained heterogeneity. We show that the consumer’s environment affects risk aversion. Individuals who are more likely to face income uncertainty or to become liquidity constrained exhibit a higher degree of absolute risk aversion, consistent with recent theories of attitudes toward risk in the presence of uninsurable risks. (JEL: D1, D8) 1. Introduction The relationship between wealth and a consumer’s attitude toward risk—as indi- cated, for instance, by the degree of absolute risk aversion or of absolute risk tolerance—is central to many fields of economics. As argued by Kenneth Arrow more than 35 years ago, “the behavior of these measures as wealth varies is of the greatest importance for prediction of economic reactions in the presence of uncertainty” (1970, p. 35). The editor in charge of this paper was Jordi Galí Acknowledgments: We thank Chris Carroll, Pierre-André Chiappori, Christian Gollier, Michaelis Haliassos, Winfried Koeniger, Wilko Letterie, John Pratt, Andrea Tiseno, and Richard Zeckhauser for very valuable discussion and suggestions.
    [Show full text]
  • Chapter 2 — Risk and Risk Aversion
    Chapter 2 — Risk and Risk Aversion The previous chapter discussed risk and introduced the notion of risk aversion. This chapter examines those concepts in more detail. In particular, it answers questions like, when can we say that one prospect is riskier than another or one agent is more averse to risk than another? How is risk related to statistical measures like variance? Risk aversion is important in finance because it determines how much of a given risk a person is willing to take on. Consider an investor who might wish to commit some funds to a risk project. Suppose that each dollar invested will return x dollars. Ignoring time and any other prospects for investing, the optimal decision for an agent with wealth w is to choose the amount k to invest to maximize [uw (+− kx( 1) )]. The marginal benefit of increasing investment gives the first order condition ∂u 0 = = u′( w +− kx( 1)) ( x − 1) (1) ∂k At k = 0, the marginal benefit isuw′( ) [ x − 1] which is positive for all increasing utility functions provided the risk has a better than fair return, paying back on average more than one dollar for each dollar committed. Therefore, all risk averse agents who prefer more to less should be willing to take on some amount of the risk. How big a position they might take depends on their risk aversion. Someone who is risk neutral, with constant marginal utility, would willingly take an unlimited position, because the right-hand side of (1) remains positive at any chosen level for k. For a risk-averse agent, marginal utility declines as prospects improves so there will be a finite optimum.
    [Show full text]
  • Single-Period Portfolio Choice and Asset Pricing
    Part I Single-Period Portfolio Choice and Asset Pricing 1 Chapter 1 Expected Utility and Risk Aversion Asset prices are determined by investors’ risk preferences and by the distrib- utions of assets’ risky future payments. Economists refer to these two bases of prices as investor "tastes" and the economy’s "technologies" for generating asset returns. A satisfactory theory of asset valuation must consider how in- dividuals allocate their wealth among assets having different future payments. This chapter explores the development of expected utility theory, the standard approach for modeling investor choices over risky assets. We first analyze the conditions that an individual’spreferences must satisfy to be consistent with an expected utility function. We then consider the link between utility and risk aversion and how risk aversion leads to risk premia for particular assets. Our final topic examines how risk aversion affects an individual’s choice between a risky and a risk-free asset. Modeling investor choices with expected utility functions is widely used. However, significant empirical and experimental evidence has indicated that 3 4 CHAPTER 1. EXPECTED UTILITY AND RISK AVERSION individuals sometimes behave in ways inconsistent with standard forms of ex- pected utility. These findings have motivated a search for improved models of investor preferences. Theoretical innovations both within and outside the expected utility paradigm are being developed, and examples of such advances are presented in later chapters of this book. 1.1 Preferences when Returns Are Uncertain Economists typically analyze the price of a good or service by modeling the nature of its supply and demand. A similar approach can be taken to price an asset.
    [Show full text]
  • An Approximate Solution for the Power Utility Optimization Under Predictable Returns Arxiv:1911.06552V3 [Q-Fin.PM] 21 Jan 2021
    An approximate solution for the power utility optimization under predictable returns Dmytro Ivasiuk Department of Statistics, European University Viadrina, Frankfurt(Oder), Germany E-mail: [email protected] Abstract This work presents an approximate single period solution of the portfolio choice problem for the investor with a power utility function and the predictable returns. Assuming that gross portfolio returns approximately follow a log-normal distribu- tion, it was possible to derive the optimal weights. As it is known, the log-normal distribution seems to be a good proxy of the normal distribution in case if the stan- dard deviation of the last one is way much smaller than its mean. Thus, succeeding to model the behaviour of asset returns as a multivariate normal (conditional nor- mal) distribution (what is a popular way to proceed in academic literature) would be the case to apply the ideas discussed in this work. Besides, the paper provides a simulation study comparing the derived result to the well-know numerical solution. Keywords: CRRA, power utility, utility optimization, numerical solution, log-normal distribution. 1 Introduction The current paper discusses portfolio optimization as the utility of wealth maximization. The classical problem stands for deriving the value function given by the whole investment period with the use of the Bellman backward method (see Pennacchi, 2008; Brandt, 2009). One can find it as the typical solution for the investment strategies for different types of utility functions (see Bodnar et al., 2015a,b). However, most of the time, it is hard arXiv:1911.06552v3 [q-fin.PM] 21 Jan 2021 to succeed and to derive an analytical solution.
    [Show full text]