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Jensen's alpha is a statistic that is commonly used in empirical finance to assess the marginal return associated with unit exposure to a given strategy. Generalizing the above definition to the multifactor setting, Jensen's alpha is a measure of the marginal return associated with an additional strategy that is not explained by existing factors.
Jensen's inequality can be proved in several ways, and three different proofs corresponding to the different statements above will be offered. Before embarking on these mathematical derivations, however, it is worth analyzing an intuitive graphical argument based on the probabilistic case where X is a real number (see figure).
In probability theory and statistics, the Jensen–Shannon divergence, named after Johan Jensen and Claude Shannon, is a method of measuring the similarity between two probability distributions. It is also known as information radius ( IRad ) [ 1 ] [ 2 ] or total divergence to the average . [ 3 ]
Michael Cole Jensen (November 30, 1939 – April 2, 2024) was an American economist who worked in the field of financial economics. From 1967-1988, he was on the University of Rochester's faculty. [ 1 ]
This follows immediately from Jensen’s inequality: ) = () ... as alpha varies from -1 to 2. ... Financial interpretation
The information ratio is often annualized. While it is then common for the numerator to be calculated as the arithmetic difference between the annualized portfolio return and the annualized benchmark return, this is an approximation because the annualization of an arithmetic difference between terms is not the arithmetic difference of the annualized terms. [6]
Since the square root is a strictly concave function, it follows from Jensen's inequality that the square root of the sample variance is an underestimate. The use of n − 1 instead of n in the formula for the sample variance is known as Bessel's correction , which corrects the bias in the estimation of the population variance, and some, but ...
The Fisher metric also allows the action and the curve length to be related to the Jensen–Shannon divergence. [7] Specifically, one has ( b − a ) ∫ a b ∂ θ j ∂ t g j k ∂ θ k ∂ t d t = 8 ∫ a b d J S D {\displaystyle (b-a)\int _{a}^{b}{\frac {\partial \theta ^{j}}{\partial t}}g_{jk}{\frac {\partial \theta ^{k}}{\partial t}}\,dt ...