Lars Tyge Nielsen

Equilibrium in CAPM without a riskless asset

In the mean-variance CAPM without a riskless asset, the possibility of satiation sometimes leads to non-existence of general equilibrium. Moreover, because portfolio preferences are not necessarily monotone, equilibrium asset prices, when they exist, may be negative or zero. To demonstrate the possibility of non-existence, and to develop an intuitive understanding of when and why equilibrium does or does not exist, this paper fully investigates the special case of utility functions linear in mean and variance and partially extends the results to the general case.

The Instantaneous Capital Market Line

We show that if the intercept and slope of the instantaneous capital market line are deterministic, then investors will not hold any hedge portfolios in the sense of Merton [9, 11]. They will choose portfolios that plot on the capital market line, and they will slide up and down the capital market line over time as their wealth and risk tolerance change. This result allows us to aggregate over investors and derive a single factor CAPM where the first and second moments of security returns may change stochastically over time and markets are potentially incomplete. Copyright Springer-Verlag Berlin/Heidelberg 2006

The utility of infinite menus

Abstract It is sometimes concluded from the St. Petersburg paradox that von Neumann-Morgenstern utility functions must be bounded. Some axiom systems for expected utility avoid this conclusion and allow for abounded utility functions. This note goes a step further and constructs an axiom system which may yield a utility function that takes the value plus infinity. Such a utility function is potentially applicable in situations where ‘blank checks’ or ‘infinite menus’ are ranked along with other prospects.

The expected utility of portfolios of assets

Abstract This paper describes two models of asset markets and portfolio choice: one where the von Neumann-Morgenstern utility function is defined on the non-negative real line and short-selling is not allowed, and one where the von Neumann-Morgenstern utility function is defined on the entire real line and short-selling may be possible. A number of properties of the derived utility function for portfolios, needed in demand and equilibrium analysis, are investigated with particular attention to the possibility that the von Neumann-Morgenstern utility function may be unbounded below and that the derived (expected) utility of some portfolios may be negative infinity.

Monotone Risk Aversion

This paper defines decreasing absolute risk aversion in purely behavioral terms without any assumption of differentiability and shows that a strictly increasing and risk averse utility function with decreasing absolute risk aversion is necessarily differentiable with an absolutely continuous derivative. A risk averse utility function has decreasing absolute risk aversion if and only if it has a decreasing absolute risk aversion density, and if and only if the cumulative absolute risk aversion function is increasing and concave. This leads to a characterization of all such utility functions. Analogues of these results also hold for increasing absolute and for increasing and decreasing relative risk aversion.

Common knowledge: The case of linear regression

Abstract This paper reinterprets the multivariate McKelvery-Page theorem as a special case of a result about orthogonal projections in a Hilbert space. When information is given by linear signals and the distribution of payoffs and signals is elliptical, it is shown how common and pooled information can be represented by information matrices, and the multivariate mcKelvey-Page theorem is reinterpreted as a result about matrix algebra. Applied to a version of Grossmans securities market model with asymmetric information, the result implies that the equilibrium price is common knowledge only if all investors agree on their conditional expectations of payoffs. Combined with a result about pooling of linear signals, this observation implies that the linear rational expectations equilibrium is unique.

Robustness of the market model

SummaryThe market model specifies that the random vector of returns on risky assets is an affine function of the return on the market portfolio plus a residual which has zero conditional expectation given the return on the market. The model is important because of its intimate relation to distributional two-fund separation and the CAPM equation. This paper shows that the market model is robust to small changes in the asset supplies only if the distribution of returns is spherically generated.

Pareto optima in incomplete financial markets

Abstract This paper demonstrates the structure of the set of Pareto optima in exchange economy models where the choice sets are unbounded and satiation is possible. The main examples are found among asset market models with short-selling and incomplete markets, such as the mean–variance Capital Asset Pricing Model (CAPM) without a riskless asset.