1,594 research outputs found
Optimal leverage from non-ergodicity
In modern portfolio theory, the balancing of expected returns on investments
against uncertainties in those returns is aided by the use of utility
functions. The Kelly criterion offers another approach, rooted in information
theory, that always implies logarithmic utility. The two approaches seem
incompatible, too loosely or too tightly constraining investors' risk
preferences, from their respective perspectives. The conflict can be understood
on the basis that the multiplicative models used in both approaches are
non-ergodic which leads to ensemble-average returns differing from time-average
returns in single realizations. The classic treatments, from the very beginning
of probability theory, use ensemble-averages, whereas the Kelly-result is
obtained by considering time-averages. Maximizing the time-average growth rates
for an investment defines an optimal leverage, whereas growth rates derived
from ensemble-average returns depend linearly on leverage. The latter measure
can thus incentivize investors to maximize leverage, which is detrimental to
time-average growth and overall market stability. The Sharpe ratio is
insensitive to leverage. Its relation to optimal leverage is discussed. A
better understanding of the significance of time-irreversibility and
non-ergodicity and the resulting bounds on leverage may help policy makers in
reshaping financial risk controls.Comment: 17 pages, 3 figures. Updated figures and extended discussion of
ergodicit
Seven Sins in Portfolio Optimization
Although modern portfolio theory has been in existence for over 60 years,
fund managers often struggle to get its models to produce reliable portfolio
allocations without strongly constraining the decision vector by tight bands of
strategic allocation targets. The two main root causes to this problem are
inadequate parameter estimation and numerical artifacts. When both obstacles
are overcome, portfolio models yield excellent allocations. In this paper,
which is primarily aimed at practitioners, we discuss the most common mistakes
in setting up portfolio models and in solving them algorithmically
On the Equivalence of Quadratic Optimization Problems Commonly Used in Portfolio Theory
In the paper, we consider three quadratic optimization problems which are
frequently applied in portfolio theory, i.e, the Markowitz mean-variance
problem as well as the problems based on the mean-variance utility function and
the quadratic utility.Conditions are derived under which the solutions of these
three optimization procedures coincide and are lying on the efficient frontier,
the set of mean-variance optimal portfolios. It is shown that the solutions of
the Markowitz optimization problem and the quadratic utility problem are not
always mean-variance efficient. The conditions for the mean-variance efficiency
of the solutions depend on the unknown parameters of the asset returns. We deal
with the problem of parameter uncertainty in detail and derive the
probabilities that the estimated solutions of the Markowitz problem and the
quadratic utility problem are mean-variance efficient. Because these
probabilities deviate from one the above mentioned quadratic optimization
problems are not stochastically equivalent. The obtained results are
illustrated by an empirical study.Comment: Revised preprint. To appear in European Journal of Operational
Research. Contains 18 pages, 6 figure
Management Science, Economics and Finance: A Connection
This paper provides a brief review of the connecting literature in management science, economics and finance, and discusses some research that is related to the three disciplines. Academics could develop theoretical models and subsequent econometric models to estimate the parameters in the associated models, and analyze some interesting issues in the three disciplines
Arbitrage Portfolios
It should be expected from this paper an expansion on some distinctive issues regarding arbitrage portfolios: i) a definition on arbitrage portfolios that enables adjustments to SML and CML environments; ii) sufficient conditions to set up arbitrage portfolios against the SML and CML; iii) feasibility of separation portfolios to carry out arbitrage not only against SML but CML as well; iv) arbitrage of portfolios located in Treynor’s lines by using separation portfolios within a SML environment.
Mean-Variance and Expected Utility: The Borch Paradox
The model of rational decision-making in most of economics and statistics is
expected utility theory (EU) axiomatised by von Neumann and Morgenstern, Savage
and others. This is less the case, however, in financial economics and
mathematical finance, where investment decisions are commonly based on the
methods of mean-variance (MV) introduced in the 1950s by Markowitz. Under the
MV framework, each available investment opportunity ("asset") or portfolio is
represented in just two dimensions by the ex ante mean and standard deviation
of the financial return anticipated from that investment.
Utility adherents consider that in general MV methods are logically incoherent.
Most famously, Norwegian insurance theorist Borch presented a proof suggesting
that two-dimensional MV indifference curves cannot represent the preferences of
a rational investor (he claimed that MV indifference curves "do not exist").
This is known as Borch's paradox and gave rise to an important but generally
little-known philosophical literature relating MV to EU. We examine the main
early contributions to this literature, focussing on Borch's logic and the
arguments by which it has been set aside.Comment: Published in at http://dx.doi.org/10.1214/12-STS408 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Portfolio Selection in Incomplete Markets with Utility Maximisation
The problem of maximizing the expected utility is well understood in the context of a complete financial market. This dissertation studies the same problem in an arbitrage-free yet incomplete market. Jin and Zhou have characterized the set of the terminal wealths that can be replicated by admissible portfolios. The problem is then transformed into a static optimization problem. It is proved that the terminal wealth is attainable for all utility functions when the market parameters are deterministic. The optimal portfolio is obtained explicitly when the utility function is logarithmic even if the market parameters follow stochastic processes. However we do not succeed in extending this result to the power utility function
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