  In this paper we develop a novel methodology for estimation of risk capital
allocation. The methodology is rooted in the theory of risk measures. We work
within a general, but tractable class of law-invariant coherent risk measures,
with a particular focus on expected shortfall. We introduce the concept of fair
capital allocations and provide explicit formulae for fair capital allocations
in case when the constituents of the risky portfolio are jointly normally
distributed. The main focus of the paper is on the problem of approximating
fair portfolio allocations in the case of not fully known law of the portfolio
constituents. We define and study the concepts of fair allocation estimators
and asymptotically fair allocation estimators. A substantial part of our study
is devoted to the problem of estimating fair risk allocations for expected
shortfall. We study this problem under normality as well as in a nonparametric
setup. We derive several estimators, and prove their fairness and/or asymptotic
fairness. Last, but not least, we propose two backtesting methodologies that
are oriented at assessing the performance of the allocation estimation
procedure. The paper closes with a substantial numerical study of the subject.
