Michael J. Stutzer

An Information-Theoretic Alternative to Generalized Method of Moments Estimation

While optimally weighted generalized method of moments (GAM) estimation has desirable large sample properties, its small sample performance is poor in some applications. The authors propose a computationally simple alternative, for weakly dependent data generating mechanisms, based on minimization of the Kullback-Leibler information criterion. Conditions are derived under which the large sample properties of this estimator are similar to GAM, i.e., the estimator will be consistent and asymptotically normal, with the same asymptotic covariance matrix as GAM. In addition, the authors propose overidentifying and parametric restrictions tests as alternatives to analogous GAM procedures.

ADVERSE SELECTION, AGGREGATE UNCERTAINTY, AND THE ROLE FOR MUTUAL INSURANCE COMPANIES

A model of insurance markets analogous to that of Rothschild and Stiglitz (1976) is considered, with the additional feature that risk-neutral insurers face aggregate (undiversifiable) risk. In the presence of adverse selection and aggregate uncertainty and under a simple nondegeneracy condition on loss probabilities, the authors show that any equilibrium has the feature that some agents purchase participating policies (from mutual insurers) while others purchase nonparticipating policies (and, hence, do not share risk with their insurer). Specifically, agents with low loss probabilities signal their type by sharing aggregate risks with their insurer. Some empirical support for this prediction is provided. Copyright 1990 by the University of Chicago.

Portfolio Choice with Endogenous Utility: A Large Deviations Approach

This paper provides an alternative behavioral foundation for an investors use of power utility in the objective function and its particular risk aversion parameter. The foundation is grounded in an investors desire to minimize the objective probability that the growth rate of invested wealth will not exceed an investor-selected target growth rate. Large deviations theory is used to show that this is equivalent to using power utility, with an argument that depends on the investors target, and a risk aversion parameter determined by maximization. As a result, an investors risk aversion parameter is not independent of the investment opportunity set, contrary to the standard model assumption.

A Bayesian approach to diagnosis of asset pricing models

A large literature has arisen which exploits a particular portfolio on the mean-variance frontier, determining a minimum variance bound on the set of stochastic discount factors (state price to probability ratios). This paper proposes a new variational characterization of the closely related set of state price densities, based on minimization of the Kullback-Leibler Information Criterion. In contrast to the variance bound, the resulting information bound automatically satisfies an important positivity constraint. Furthermore, the information bound is determined by a portfolio which maximizes expected CARA utility. Several interpretations of the information bound are given, and empirical uses of it are illustrated.

Credit Rationing and Government Loan Programs: A Welfare Analysis

Asymmetric information about borrower default probabilities may lead to inefficient credit rationing of low-risk borrowers in otherwise competitive markets. In a simple model having these properties, we show that some types of government loan programs, such as loan guarantees issued through lenders, might improve economic efficiency. But the incentive for high-risk borrowers to misrepresent their loan quality is worsened by other government loan programs, notably those that try to target aid directly to rationed borrowers. As such, cost-effective programs may increase inefficiency. This surprising result highlights the need to conduct model-specific policy analyses, as opposed to analyses based on model-free performance indicators. Copyright American Real Estate and Urban Economics Association.

Connections Between Entropic and Linear Projections in Asset Pricing Estimation

The concept of entropy has a long and distinguished history in the physical sciences and engineering, in fields ranging from thermodynamics to image processing. Each of these applications employs a probability distribution that solves a relative entropy projection problem, i.e. an optimization problem with an entropy objective, subject to linear (e.g. moment) constraints. This paper develops the relationship between relative entropy projection approaches and the better-known linear projection approaches to problems of estimation and performance diagnostics for stochastic discount factor models in asset pricing. Frequentist interpretations of relative entropy, enabled by large deviations theory, are used to unify the interpretation of the seemingly disparate procedures.

Simple Entropic Derivation of a Generalized Black-Scholes Option Pricing Model

A straightforward derivation of the celebrated Black-Scholes Option Pricing model is obtained by solution of a simple constrained minimization of relative entropy. The derivation leads to a natural generalization of it, which is consistent with some evidence from stock index option markets.

Die Geographie der Kreativen Klasse in Deutschland

KurzfassungDer Beitrag analysiert die räumliche Verteilung verschiedener Kategorien kreativer Personen in Deutschland. Allgemein ist der Anteil der Kreativen in den Städten höher als auf dem Land. Ein etwas abweichendes Standortverhalten zeigen die freiberuflichen Künstler, die auch in einigen ländlichen Regionen stärker vertreten sind. Ein hoher Anteil an Kreativen in einer Region kann mit einem hohen Niveau der öffentlichen Versorgung und einem hohen Ausländeranteil als Indikator für die „Offenheit” eines Milieus erklärt werden. Gute Beschäftigungschancen haben nur einen schwachen Einfluss. Regionen mit einem hohen Anteil an Kreativen sind durch ein relativ hohes Niveau an Gründungen, Innovationen und einen hohen Anteil an Beschäftigten in Hightech-Branchen gekennzeichnet.AbstractThe article analyses the regional distribution of different categories of creative individuals in Germany. Generally, the share of creative people is higher in cities as compared to the rural areas. The freelancing artists are a kind of exception in this respect; they constitute a relatively high share of the population in some rural areas. A high share of creative people in a region can be explained by a high level of public provisions and a high share of foreign born population, which can be regarded as an indicator of the “openness” in the local milieu. Good employment opportunities have only a relatively weak impact. Regions with a high share of creatives tend to have an above average level of new business formation, a high level of innovation and a relatively high share of employees in high-tech industries.

Asset Allocation without Unobservable Parameters

Some asset allocation advice for long-term investors is based on maximization of expected utility. Most commonly used investor utilities require measurement of a risk-aversion parameter appropriate to the particular investor. But accurate assessment of this parameter is problematic at best. Maximization of expected utility is thus not only conceptually difficult for clients to understand but also difficult to implement. Other asset allocation advice is based on minimizing the probability of falling short of a particular investors long-term return target or of an investable benchmark. This approach is easier to explain and implement, but it has been criticized by advocates of expected utility. These seemingly disparate criteria can be reconciled by measuring portfolio returns relative to the target (or benchmark) and then eliminating the usual assumption that the utilitys risk-aversion parameter is not also determined by maximization of expected utility. Financial advisors should not be persuaded by advocates of the usual expected-utility approach. Asset allocation advice for long-term investors is based on a variety of criteria. Some advice is based on maximization of expected utility. The most commonly used utility functions are (1) quadratic or exponential, which yield the ubiquitous mean-variance utility underlying modern portfolio theory, and (2) the constant relative risk-aversion (CRRA) power utility. Both utilities require measurement of a risk-aversion parameter appropriate to a particular investor. But no validated procedures exist for reliably assessing an individuals risk-aversion parameter, and some investigators have suggested that all such procedures are doomed to failure because the risk aversion of an individual can depend on the scale of risks encountered. Other asset allocation advice for long-term investors is based on a different criterion: minimizing the probability of falling short of a particular investors targeted long-term return or an investable benchmark. This approach is grounded in the findings of behavioral finance, may be easier to explain to investors than maximization of expected utility, and obviates the need to assess a risk-aversion parameter. The article presents a description of the two criteria and illustrates specifically how to implement each in the three-step asset-allocation process: (1) choosing a criterion function to maximize, (2) using historical time-series (or some other) data on asset class returns to estimate optimal asset allocations consistent with the chosen criterion function, and (3) using specific investor information to select the asset allocation appropriate for the particular investor. Then, I argue that the CRRA-utility and shortfall-probability analyses can be reconciled. Surprisingly, the seemingly disparate conventional CRRA-utility-maximization and shortfall-probability-minimization methods can be reconciled by completely maximizing the expected CRRA utility of the ratio of the portfolios return to the investors target return. This maximization requires unconventionally maximizing the expected utility by selection of both the portfolios asset allocation weights and the utilitys risk-aversion parameter (as opposed to conventional maximization over the weights alone with the use of some fixed value of the risk-aversion parameter). This unconventional formulation of minimizing long-run target shortfall probability retains the framework of expected-utility maximization while eliminating the conventional but problematic requirement that the advisor fix a value of the risk-aversion parameter that is most appropriate for the investor. Instead, in an interactive feedback process, the advisor and the investor mutually determine the most appropriate target rate of return. I use a simple two-asset allocation problem to illustrate this approach. The results are quite sensible and lead to a reexamination of the arguments put forth by advocates of the conventional use of expected utility and of the arguments against the minimization of shortfall probability. Criticisms of the use of shortfall probability are either overstated or not applicable to target-shortfall minimization (target-outperformance maximization) as described in this article. Theorists who believe that this criterion is inferior to risk aversion parameter-dependent expected utility need to reevaluate that position in light of the implementation and the risk-scaling problems highlighted in this article.

The Misuse of Expected Returns

Much textbook emphasis is placed on the mathematical notion of expected return and its historical estimate via an arithmetic average of past returns. But those wanting to forecast a typical future cumulative return should be more interested in estimating the median future cumulative return than in estimating the mathematical expected cumulative return. For that purpose, continuous compounding of the mathematical expected log gross return is more relevant than ordinary compounding of the mathematical expected gross return. Self-test Pensions, endowments, and other long-term investors often want to forecast the future cumulative returns associated with various asset-class indices or investment strategies. Because no one can foretell the future, the future cumulative return is always a random variable that has a probability distribution. As a point forecast of the future cumulative return, some analysts have chosen to estimate the mathematical expectation of the future cumulative return’s distribution. We argue that this choice is misguided because the distribution of the future cumulative return is often heavily (positively) skewed. As a result, the mathematical expectation of its distribution is not as good a measure of its central tendency (i.e., what is more likely to happen) as is the median future cumulative return. The median future cumulative return has a 50 percent chance of being met or exceeded, but we show that the probability of meeting or exceeding the mathematical expectation approaches zero as the forecast horizon grows to infinity. As a result, even an accurate forecast of the mathematical expected future cumulative return is a badly overoptimistic forecast of what is likely to occur over long horizons. For example, our simulations indicate that there is only about a 30 percent probability of meeting or exceeding the mathematical expected future cumulative return of a large-capitalization stock index at the 30-year horizon that typifies retirement planning forecasts. We use a relatively recent result in the theory of statistics to argue that analysts who want to estimate the median future cumulative return should focus their attention on the mathematical expected logarithm of a single period’s gross return distribution. Continuously compounding the expected log gross return through T periods approximates the median future cumulative return at the T-period horizon. A simple point forecast of the median future cumulative return is made by (1) computing the average of the historical log gross returns (e.g., historical daily or monthly return data) in all past measurement periods and then (2) continuously compounding Step 1’s result up to the T-period forecast horizon. Substituting the average historical ordinary net return for the average historical log gross return in Step 1 is not recommended. Unfortunately, use of any historical average return is somewhat problematic, even in ideal statistical circumstances that may not characterize the real world. Even if the distribution of period log gross returns has been (and will remain) stable over time, the volatility of these log gross returns can make historical averages significantly different from future long-term averages. We show that typical stock index return volatility (15 percent) is enough to cause substantial fluctuation in historical averages. For example, even with 54 years of historical log gross return data, the fluctuation in future historical log gross return averages will be ±400 bps.