In this paper, the expectation of the reciprocal of first-degree polynomials of non-negative valued random variables is calculated. This is motivated to compute the Kelly criterion, which is the optimal solution of the maximization of the expected logarithm of the investment return. As soon as the expectation of the reciprocal of first-degree polynomials of asset returns is calculated, which is our main interest, the Kelly criterion can be obtained by using the ordinary optimization technique or applying the appropriate algorithm.
This paper develops a model appeared in the literature whose focus was the way rational risk averse investors anticipate the correlation breakdowns of asset returns in periods of excess demand. That model analysed the dynamics of the “expected” returns of the risky asset, and their consistency with empirical evidence. However, the same model did not provide any evidence on actual correlation generated by the dynamics of returns. A model to link asset returns to excess demand is required to analyse the implied correlation between the securities traded. In this work we estimate such a model. Results confirm that the expected and ex-post correlation tend to move closely. In other words a self-fulfilling prophecy about correlation breakdown can take place, even when rational agents dominate the financial market.
The paper is devoted to the investigation of statistical properties of the sample estimator of the beta coefficient in the case when the weights of benchmark portfolio are constant and for the target portfolio, the global minimum variance portfolio is taken. We provide the asymptotic distribution of the sample estimator of the beta coefficient assuming that the asset returns are multivariate normally distributed. Based on the asymptotic distribution we construct the confidence interval for the beta coefficient. We use the daily returns on the assets included in the DAX index for the period from 01.01.2018 to 30.09.2019 to compare empirical and asymptotic means, variances and densities of the standardized estimator for the beta coefficient. We obtain that the bias of the sample estimator converges to zero very slowly for a large number of assets in the portfolio. We present the adjusted estimator of the beta coefficient for which convergence of the empirical variances to the asymptotic ones is not significantly slower than for a sample estimator but the bias of the adjusted estimator is significantly smaller.
On Calculating Method of the Kelly Criterion for Financial Investment in Single Risky Asset with Various Distributions of Returns