27 research outputs found
Analysis of Kelly-optimal portfolios
We investigate the use of Kelly's strategy in the construction of an optimal
portfolio of assets. For lognormally distributed asset returns, we derive
approximate analytical results for the optimal investment fractions in various
settings. We show that when mean returns and volatilities of the assets are
small and there is no risk-free asset, the Kelly-optimal portfolio lies on
Markowitz Efficient Frontier. Since in the investigated case the Kelly approach
forbids short positions and borrowing, often only a small fraction of the
available assets is included in the Kelly-optimal portfolio. This phenomenon,
that we call condensation, is studied analytically in various model scenarios.Comment: 15 pages, 7 figures; extended list of references and some minor
modification
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