Günter Bamberg

Implications of Constant Risk Aversion

The assumption of constant risk aversion often leads to a considerable simplification of decision theoretic analyses. It is shown that this restriction on constant risk aversion permits the description of a wide range of risk averse patterns between the extreme cases of risk neutrality and the exclusive orientation on the pessimistic maxmin criterion. In particular, a new axiomatic foundation of the constant risk aversion is given and a series of properties for the certainty equivalent are derived. Possible applications are shown by the treatment of risk premiums, values of information, portfolio selection, capital market theory and syndicates.ZusammenfassungDie Zugrundelegung einer konstanten Risikoaversion vereinfacht entscheidungstheoretische Analysen oft erheblich. In dieser Arbeit wird gezeigt, daß die Beschränkung auf konstante Risikoaversion ein großes Spektrum risikoaverser Verhaltensweisen zwischen den Extremfällen der Risikoneutralität und der alleinigen Orientierung am pessimistischen Maximin-Kriterium zu beschreiben gestattet. Insbesondere werden eine neue axiomatische Begründung der konstanten Risikoaversion vorgestellt und eine Reihe von Eigenschaften für das zugehörige Sicherheitsäquivalent hergeleitet. Zur Verdeutlichung der Anwendungsmöglichkeiten werden exemplarisch Risikoprämien, Informationswerte, Portfolio-Selektion und Kapitalmarkttheorie sowie Syndikate behandelt.

Capital market equilibria

Prologue.- 1. Equilibrium versus Market Imperfections.- 2. Questions and Answers.- The Hybrid Model and Related Approaches to Capital Market Equilibria.- 1. Introduction.- 2. Portfolio Models Based on Different Sets of Parameters.- 2.1 One-Parameter Models.- 2.2 Two-Parameter Models: Mean-Semivariance Approach.- 2.3 Other Two-Parameter Approaches.- 2.4 Extensions to Three or More Parameters.- 3. Rationale of the Hybrid Model.- 3.1 Consistency of the Mean-Variance Approach with Expected Utility and Stochastic Dominance.- 3.2 Explicit Solutions of the Portfolio Problem.- 3.3 Explicit Solutions of the Equilibrium Conditions.- 3.4 Which Mean-Variance Approaches Provide Explicit Solutions?.- 4. Applications of the Hybrid Model.- 4.1 Consideration of Income Taxation.- 4.2 Heterogeneous Expectations.- 4.3 Restrictions on Short Sales.- 4.4 Some Other Market Imperfections.- 5. Appendix.- 5.1 Proof of Theorem 4.- 5.2 Solution of Partial Differential Equation (31).- References.- Portfolio Decisions and Capital Market Equilibria Under Incomplete Information.- 1. Introduction.- 2. Risk Situation with Regard to the Prior Parameters: A Two-Level Bayes Approach.- 3. Risk Situation with Regard to the Prior Parameters: Lins Approach.- 4. Partial Uncertainty with Regard to the Prior Parameters.- 5. Asset Pricing under Uncertainty.- References.- Option Valuation: Theory and Empirical Evidence.- 1. Introduction.- 2. Option Valuation Theory.- 2.1 Preference and Distribution-Free Results.- 2.1.1 Call Options.- 2.1.2 Put Options.- 2.1.3 Relations Between Puts and Calls.- 2.1.4 Additional Arbitrage Restrictions.- 2.2 Distributional Assumptions and Hedging Models.- 2.2.1 Hedge Portfolios.- 2.2.2 The Classical Black-Scholes Model.- 2.2.3 A Brief Description of Other Option Valuation Models.- 2.2.4 Analytic Models For American Calls and Puts.- 2.3 Preference Assumptions and Non-Hedging Models.- 2.4 New Option Instruments.- 2.5 Applications of Option Theory.- 3. Empirical Tests of Option Valuation.- 3.1 Test of Boundary Conditions Among an Individual Equity Option and the Underlying Stock.- 3.2 Test of Boundary Conditions Among Different Equity Options and the Underlying Stock.- 3.3 Tests of Equity Option Pricing Models.- 3.3.1 Results of Robustness Tests.- 3.3.2 Results of Predictability Tests.- 3.3.3 Results of Unbiasedness Tests.- 3.3.4 Results of Hedge Return Behavior Tests.- 3.4 Tests of New Option Instruments.- 3.5 Estimation Problems.- 4. Appendix: Formulae for the Evaluation of European Calls.- References.- The Value of Security Agreements.- 1. A Survey of Credit Support Decision Models.- 1.1 Credit Decisions in a Restricted Capital Market.- 1.2 Market Uncertainty.- 1.3 Credit Support Decisions with Event Uncertainty.- 2. Neoclassical Theory and Secured Debt.- 2.1 The Basic Approach.- 2.2 Secured Debt and Capital Market Equilibrium.- 2.3 Collateral Policy and Non-Market-Value Debt Claims.- 3. The Theory of Credit Support Decisions in the Light of the Economics of Information.- 3.1 Collaterals as a Tool to Limit the Creditability Risk.- 3.2 Contemporaneous Examination of Credit Standing Risk and Credit Reliability Risk.- 3.2.1 Changing the Dividend Policy.- 3.2.2 Changing the Credit Policy.- 3.2.3 Changing the Investment Policy.- 4. A Scheme of Credit Contract Covenants.- 4.1 Credit Contract Covenants.- 4.2 Covenants Referring Indirectly to Means of Payment.- 4.2.1 Special Obligations of the Debtors.- 4.2.2 Special Rights of the Creditors.- 4.3 Claims of Creditors which Refer to Means of Payment.- 5. The Efficiency of Securing Debt.- 6. Appendix: Secured Debt and Uncertainty.- 6.1 The Firms Position.- 6.2 Derivation of the Value of Secured Debt.- 6.3 Market Value of the Debt in Dependence of its Collateral Policy.- 6.4 The Maximization of the Market Value with Total Collateral Policy.- References.- Asset Pricing in a Small Economy: A Test of the Omitted Assets Model.- 1. Introduction.- 2. Portfolio Based Tests of Efficiency.- 3. Omitted Assets.- 3.1 The Model.- 3.2 Integrated and Segmented Capital Markets.- 4. The Data.- 5. Efficiency of the Canadian Market Index.- 5.1 Beta Estimation under Thin Trading.- 5.2 Preliminary Estimates.- 5.3 Maximum Likelihood Estimation.- 6. Tests of the Omitted Assets Hypothesis.- Appendix: Iterative Maximum Likelihood Procedure.- References.- The Simple Analytics of Arbitrage.- 1. General Equilibrium, Modern Finance, and Arbitrage Theory.- 1.1 General Equilibrium.- 1.2 Modern Finance.- 1.3 Arbitrage.- 2. An Example, its Generalization, and the Question.- 2.1 A Portfolio of Savings Bonds and Term Deposits.- 2.2 Is a Free Lunch Possible?.- 2.3 Events and Futures.- 2.4 The Question: Should the Original Portfolio be Revised?.- 3. Arbitrage versus Equilibrium.- 3.1 Some Borrowing from Production Theory.- 3.2 Either Arbitrage is Possible or Equilibrium Prices Exist.- 3.3 The Lemma of Minkowski-Farkas as Mathematical Background.- 3.4 The Derivation of an Issue Price.- 3.5 Summing up the Results.- 4. Derivative Contracts in Complete Capital Markets.- 4.1 Complete Capital Market.- 4.2 One-Period Option Pricing.- References.- About Contributors.- Author Index.

The Hybrid Model and Related Approaches to Capital Market Equilibria

Single period capital-asset-pricing models provide a number of implications on capital market equilibria. Familiar notions as, for instance, the market price of risk or the market portfolio can be derived from the equilibrium conditions without having explicit representations of the equilibrium prices. However, such explicit solutions are required to conduct other investigations successfully. The availability of explicit solutions facilitates the analysis of changes in the value of firms under a variety of circumstances. The paper concentrates on the chances to obtain explicit solutions. It makes precise in which sense the hybrid model, characterized by multivariate normally distributed returns and constant risk aversion of the investors, is the most convenient model of capital market equilibrium. Some applications are surveyed concerning taxes, heterogeneous expectations, short sales restrictions, and imperfect capital markets.

Equity Index Replication with Standard and Robust Regression Estimators

Abstract. Approximate equity index replication, based on a linear regression setting, is critically reviewed. It is shown that tracking a performance index like the German DAX necessarily leads to violations of basic assumptions of the classical regression model. Violations occur even if the model is formulated in terms of stock price levels. When the model is based on discretely or continuously compounded returns the situation is more critical. Due to these violations, the optimality properties of the regression estimators are generally weak. In the time series context, outliers in financial time series may additionally affect the standard least squares estimator. Despite of these critical points, it is argued that regression techniques may still provide a useful tool for index replication. With respect to outliers, robust estimators can potentially provide an alternative to least squares. Hence, apart from least squares, a non-redescending and a redescending robust estimator is fitted. The empirical results for the DAX are obtained with a subset portfolio containing the most heavily weighted index members. Compared to a naive weighting scheme, the results document that least squares estimation highly improves out-of-sample replication performance. Typically, the use of robust estimators does not show replication improvements. However, when substantial market movements are present in the sample, superior replication can be obtained.Zusammenfassung. Ein Performance-Index wie der DAX kann nicht durch zeitkonstante Gewichtung von Kursen dargestellt werden. Dasselbe gilt für bereinigte Kurse, falls der Beobachtungszeitraum mindestens einen Verkettungstermin umfasst. Analoge (negative) Aussagen gelten auch für diskrete Index- bzw. Aktienrenditen. Gänzlich unmöglich wird eine lineare Darstellung bei Verwendung von stetigen Renditen. Im ersten Teil der Arbeit wird aufgezeigt, daß sich diese Probleme bei Regressionsmodellen zur näherungsweisen Nachbildung des DAX in Verletzungen der Prämissen des klassischen Regressionsmodells niederschlagen. Der zweite Teil enthält einen empirischen Vergleich verschiedener Tracking-Prozeduren: naive Gleichgewichtung, Kleinst-Quadrate-Schätzung und robuste Schätzung.

On the Non-Existence of Conditional Value-at-Risk under Heavy Tails and Short Sales

Value-at-Risk (VaR) and conditional value-at-risk (CVaR) are important risk measures. Especially VaR is very popular and widespread in risk management and banking supervision. However, VaR has some unwelcome properties which are not shared by CVaR. Therefore CVaR is preferable from a theoretical point of view. Both VaR and CVaR are discussed for long and short positions. It is pointed out that short positions and heavy tails are incompatible with a finite CVaR.

Risk-Taking under Progressive Taxation: Three Partial Effects

The opinion is widely spread that taxation in general, and income taxation in particular, reduces the individual propensity to take risk. There are even sporadic statements of professional economists supporting such a view. Cf. e.g. M. Friedman (1962, p. 173). It is therefore an important theoretical insight that, by way of contrast, income taxation may encourage risk-taking. This proposition is due to Domar and Musgrave (1944) who assumed a proportional tax with full loss offset to show the effect. Domar and Musgrave’s pioneering article has stimulated a considerable number of papers to clarify the effect of taxation on risk-taking for various frameworks. On the whole the conclusion is that the effect is highly ambiguous. Its direction heavily depends on the kind of assumptions being made about the investor’s risk aversion and general tax provisions. Atkinson and Stiglitz (1980, Lecture Four) present a comprehensive analysis of various model specifications. Recent contributions consequently tend to prove indeterminacy of the effect as a generic feature instead of proving determinacy for special cases. Cf. Bamberg and Richter (1984), Bamberg (1984), Buchholz (1985).

Wahrheitsinduzierende Mechanismen, Fehlallokationen und kollusives Verhalten bei der Investitionsbudgetierung

Zur Bestimmung optimaler Investitionsbudgets in einer divisionalisierten Unternehmung ist die Zentrale auf die wahrheitsgemase Meldung der Gewinnpotentiale durch die einzelnen Divisionen angewiesen. Einen Losungsvorschlag fur dieses Problem stellt der sogenannte Groves-Mechanimus dar, unter dem die wahrheitsgemasen Meldungen immer dominante Strategien im Spiel der Divisionsmanager sind. Es wird gezeigt, das die Anreizkompatibilitat dieses Verfahrens nicht nur bei der Annahme expliziten kooperativen Verhaltens der Divisionsmanager verlorengeht, sondern das sogar stets Gleichgewichtspunkte, bestehend aus wahrheitswidrigen Meldungen, existieren, welche alle Manager besser stellen als wahrheitsgetreues Verhalten aller Beteiligten. Abschliesend wird an einem Beispiel diskutiert, inwieweit auser uberhohten Entlohnungen auch falsche Budgetierungsergebnisse resultieren konnen.

Risikobasierte Kapitalallokation in Versicherungsunternehmen unter Verwendung des Co-Semivarianz-Prinzips

Die Aufteilung des Risikokapitals einer Versicherungsunternehmung auf kleinere Einheiten (Segmente, Geschaftsbereiche, Abteilungen etc.) ist sowohl fur eine risikoadjustierte Performance-Messung als auch fur strategische Entscheidungen uber die Expansion bzw. Reduktion von Segmenten von groser Bedeutung. Verwendet man eine plausible Marginalbetrachtung und legt man die Semivarianz als Risikomas zu Grunde, erfolgt die verursachungsgerechte Allokation gemas dem Co-Semivarianz-Prinzip. Dieses Resultat wird hergeleitet und anhand von Beispielen illustriert.

Does the Planning Horizon Affect the Portfolio Structure

Does the composition of the optimal portfolio depend on the planning horizon? According to popular opinion there exists a planning horizon effect if initial wealth has to be allocated between shares and the risk-free asset: the percentage invested into shares should increase if the planning horizon is extended. The paper reviews the theoretical underpinnings of the statement. In the framework of expected utility the findings are mixed. Some results can be derived which contradict the popular opinion. But one can also find results which support the popular opinion. The conclusions depend on the class of utility functions under consideration and on the alternatives to be compared. However, from the analysis of shortfall models strong evidence in favor of the popular opinion can be inferred. In addition, the optimal percentage invested in the stock market can easily be quantified. The well-known shortfall criteria of Roy, Kataoka, and Telser are studied in some detail.

Risk Sharing and Subcontracting

Linear risk sharing provisions between companies and supply industry are considered. The provisions are characterized by target profit, target cost, and a sharing rate. The problem to assess these parameters appropriately is dealt with in a normative model. The model is parsimoniously parameterized and allows explicit solutions with respect to all contractual parameters. The simplicity of the model makes it possible to incorporate additional aspects such as diversification, heterogeneous expectations or cost monitoring expenditure.