Risk Based Capital Allocation

The present contribution reviews the procedures (absolute, incremental and marginal capital allocation) as well as the general principles (proportional allocation, covariance-principle, conditional expectation-principle, conditional value-at-risk principle, Euler-principle) for risk based capital allocation. The approaches discussed are applicable for the insurance case, the investment case and as well for credit risks.

Optimal capital allocation principles

This paper develops a unifying framework for allocating the aggregate capital of a financial firm to its business units. The approach relies on an optimisation argument, requiring that the weighted sum of measures for the deviations of the business unit’s losses from their respective allocated capitals be minimised. This enables the association of alternative allocation rules to specific decision criteria and thus provides the risk manager with flexibility to meet specific target objectives. The underlying general framework reproduces many capital allocation methods that have appeared in the literature and allows for several possible extensions. An application to an insurance market with policyholder protection is additionally provided as an illustration.Capital allocation; risk measure; comonotonicity; Euler allocation; default option; Lloyd’s of London

Capital allocation in financial institutions: the Euler method

Capital allocation is used for many purposes in financial institutions and for this purpose several methods are known. The aim of this paper is to review possible methods (we present six of them) and to help financial companies to choose between the methods. There are some properties that an allocation method should satisfy: full allocation, core compatibility, riskless allocation, symmetry and suitability for performance measurement (compatibility with Return on Risk Adjusted Capital calculation). If we think about practical application we should also consider simplicity of the methods. First we examine the methods from the point of view if they are satisfying core compatibility. We test this with simulation where we add to the existing literature that we test core compatibility with different assumptions on returns: on normal and t-distributed returns and also on returns generated from a copula. We find that if we measure risk by a coherent risk measure, the Expected Shortfall there are two methods satisfying core compatibility: the Euler method (that always fulfills the criteria) and cost gap method (obeys it around in about 99%). As Euler method is very easy to calculate even for many players while cost gap method becomes very complicated as the number of the players increases we examine further the properties of Euler method. We find that it fulfills all the above given criteria but symmetry and as aforementioned it is also very easy to calculate. Therefore we believe that the method might be suggested for practical applications.Capital Allocation, Coherent Measures of Risk, Core, Simulation

Is consumption growth consistent with intertemporal optimization? evidence from the consumer expenditure survey

In this paper we show that some of the predictions of models of consumer intertemporal optimization are in line with the patterns of nondurable expenditure observed in U.S. household-level data. We propose a flexible specification of preferences that allows multiple commodities and yields empirically tractable equations. We estimate preference parameters using the only U.S. micro data set with complete consumption information. We show that previous rejections can be explained by the simplifying assumptions made in previous studies. We also show that results obtained using good consumption or aggregate data can be misleading

Dynamic Allocation and Pricing in Incomplete Markets: A Survey

This paper surveys the recent development of empirical and theoretical researches on incomplete markets, pointing out the following aspects. First, the theoretical study in this field is motivated by empirical findings of both asset pricing anomalies and heterogeneous behavior among economic agents. Second, incomplete insurance combined with either borrowing constraints or transaction costs offers predictions consistent with empirical findings. In addition, the failure of insuring persistent or permanent shocks alone yields empirically reasonable predictions. Third, recent theoretical research has made attempts to endogenize incomplete insurance from first principles. Fourth, incomplete markets may make aggregate shocks distributed disproportionately among agents, thereby having a significant impact on dynamic allocation and pricing. Finally the theoretical research into incomplete markets triggers a reassessment of welfare implications as to business cycles, economic growth, and financial integration.

"Efficiency of Disaggregate Public Capital Provision in Japan"

We investigate the efficiency of disaggregated public capital provision for the Japanese economy. We estimate the optimality conditions based on simultaneous Euler equations by using GMM. Our results suggest that public capital productivities have been relatively high and divergent among several public capital goods. The allocation of public works is not optimal yet in Japan.

Fair Estimation of Capital Risk Allocation

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