### Conditioning The Information In Portfolio Optimization

Carlo Sala

Swiss Finance Institute at the University of Lugano

Swiss Finance Institute at the University of Lugano

October 16, 2015

Abstract:

This paper proposes a theoretical analysis on the impact of a smaller filtration set on the three main components used in asset pricing, namely the risk physical and neutral measures and the pricing kernel.

The analysis is carried out by means of a portfolio optimization problem for a small and rational investor. Solving for the maximal expected logarithmic and power utility of the terminal wealth, we prove the existence of an information premium between what is required by the theory, a complete information set arising from a fully conditional measures, and what is instead used in reality. Starting from Hansen and Jagganathan (1991) and searching for the best bounds, we study the impact of the premium on the pricing kernel. Finally, exploiting the strong interconnection between the pricing kernel and its densities the extension to the risk-neutral measure follows naturally.

### Conditioning The Information In Portfolio Optimization – Introduction

The risk-neutral measure is a risk-adjusted real world probability. Defined in a complete world with no arbitrage, the equivalence of the two measure, also known as the Equivalent Martingale Measure (EMM) follows from the second Fundamental Theorem of Asset Pricing (henceforth: FTAP). The change of measure is made possible by a kernel, an operator. In finance, once properly discounted this operator takes the name of pricing kernel (henceforth: PK). The thight interconnection between the two measures and the PK thus follows by definition. Exploiting this interrelationship, the knowledge of any of these two random variable implies uniquely the third. As a drawback, a possible misspecification in estimation may load on more variables.

The PK as well as the risk-physical measure, dealing with investors’ behaviors, are as much important as fully non linear and complex to estimate. From an estimation viewpoint, to put as much less structure as possible on the fully non-parametric PK, it is conventional wisdom to estimate separately the two measures and then extracting the relative PK from their discounted ratio. While from the point of view of the information the risk-neutral measure extracted from option panels are naturally unbiased1, the real world probabilities extracted from the underlying stream of past return are systematically biased. The backward nature of this estimation methodology is the cause of possible mispricing and is at the heart of our paper.

Investors’ subjective beliefs are forward looking. An investor decides if and how to trade depending, among the others, on her personal beliefs. Since the investment horizon spans from the present into the future it is, by definition, uncertain. This uncertainty represent the degree of riskiness of an investment. Therefore, the valuation of any risky investment, has to take into account the forward nature of the subjective beliefs.

Counterintuitively with respect to their nature, investors beliefs are estimated with backward looking information (i.e.: Ait Sahalia and Lo (1998)[1], Jackwerth (2000)[12], Brown and Jackwerth (2001)[6], Rosenberg and Engle (2002)[8], Barone-Adesi et al.(2008)[3], Hardle (2006)[20]). Using these data, an important fraction of the investor’s risk and preferences are lost. As a consequence, a discrepancy between what is empirically obtainable and what is theoretically required by the neo-classical asset pricing literature arises. The larger the forward looking information bias, the larger is the subsequent mispricing.

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