Dynamic Capital Allocation Research Articles

Dynamic capital allocation with reallocation cost

Traditional static capital allocation incurs significant reallocation costs over periods due to drastic fluctuations. We propose a new framework for dynamic capital allocation with time-varying capital constraints that incorporates reallocation costs. We obtain an analytical solution in a recursive form and demonstrate its admirable performance through simulations. The solution compromises between static allocation and expected future risk structure and strikes a balance between addressing current risk profiles and adjustment costs. It successfully provides an effective allocation tool while preserving stability.

Dynamic capital allocation rules via BSDEs: an axiomatic approach

In this paper, we study capital allocation for dynamic risk measures, with an axiomatic approach but also by exploiting the relation between risk measures and BSDEs. Although there is a wide literature on capital allocation rules in a static setting and on dynamic risk measures, only a few recent papers on capital allocation work in a dynamic setting and, moreover, those papers mainly focus on the gradient approach. To fill this gap, we then discuss new perspectives to the capital allocation problem going beyond those already existing in the literature. In particular, we introduce and investigate a general axiomatic approach to dynamic capital allocations as well as an approach suitable for risk measures induced by g-expectations under weaker assumptions than Gateaux differentiability.

Covariance Principle for Capital Allocation: A Time-Varying Approach

The covariance allocation principle is one of the most widely used capital allocation principles in practice. Risks change over time, so capital risk allocations should be time-dependent. In this paper, we propose a dynamic covariance capital allocation principle based on the variance-covariance of risks that change over time. The conditional correlation of risks is modeled by means of a dynamic conditional correlation (DCC) model. Unlike the static approach, we show that in our dynamic capital allocation setting, the distribution of risk capital allocations can be estimated, and the expected future allocations of capital can be predicted, providing a deeper understanding of the stochastic multivariate behavior of risks. The methodology presented in the paper is illustrated with an example involving the investment risk in a stock portfolio.

A note on representation of BSDE-based dynamic risk measures and dynamic capital allocations

We derive a representation for dynamic capital allocation when the underlying asset price process includes extreme random price movements. Moreover, we consider the representation of dynamic risk measures defined under Backward Stochastic Differential Equations (BSDE) with generators that grow quadratic-exponentially in the control variables. Dynamic capital allocation is derived from the differentiability of BSDEs with jumps. The results are illustrated by deriving a capital allocation representation for dynamic entropic risk measure and static coherent risk measure.

Dynamic capital allocation with irreversible investments

Capital allocation models generally assume that the risk portfolio is constructed at a single point in time, when the underwriter has full information about available underwriting opportunities. However, in practice, opportunities are not all known at the beginning but instead arrive over time. Moreover, a commitment to an opportunity is not easy to change as time passes. Thus, to optimize a portfolio, the underwriter must make decisions on opportunities as they arrive while making use of assumptions about what will arrive in the future. This paper studies capital allocation rules in this setting, finding important differences from the static setting. The pricing of an opportunity is based on an expected future marginal cost of risk associated with that opportunity—one that will be fully understood only after the risk portfolio is finalized. The risk charge for today’s opportunity is thus a probability-weighted average of the product of the marginal value of capital in future states of the world and the amount of capital consumed by the opportunity in those future states. Our numerical examples illustrate how the marginal cost of risk for an opportunity is shaped by when it arrives in time, as well as what has arrived before it.

Optimal Dynamic Capital Budgeting

Abstract I study optimal design of a dynamic capital allocation process in an organization in which the division manager with empire-building preferences privately observes the arrival and properties of investment projects, and headquarters can audit projects at a cost. Under certain conditions, a budgeting mechanism with threshold separation of financing is optimal. Headquarters: (1) allocate a spending account to the manager and replenish it over time; (2) set a threshold, such that projects below it are financed from the account, while projects above are financed fully by headquarters upon an audit. Further analysis studies when co-financing of projects is optimal and how the size of the account depends on past performance of projects.

Classical Differentiability of BSVIEs and Dynamic Capital Allocations

Capital allocations have been studied in conjunction with static risk measures in various papers. The dynamic case has been studied only in a discrete-time setting. We address the problem of allocating risk capital to subportfolios in a continuous-time dynamic context. For this purpose we introduce a classical differentiability result for backward stochastic Volterra integral equations and apply this result to derive continuous-time dynamic capital allocations. Moreover, we study a dynamic capital allocation principle that is based on backward stochastic differential equations and derive the dynamic gradient allocation for the dynamic entropic risk measure.

DIFFERENTIABILITY OF BSVIEs AND DYNAMIC CAPITAL ALLOCATIONS

Capital allocations have been studied in conjunction with static risk measures in various papers. The dynamic case has been studied only in a discrete-time setting. We address the problem of allocating risk capital to subportfolios for the first time in a continuous-time dynamic context. For this purpose, we introduce a differentiability result for backward stochastic Volterra integral equations and apply this result to derive continuous-time dynamic capital allocations. Moreover, we study a dynamic capital allocation principle that is based on backward stochastic differential equations and derive the dynamic gradient allocation for the dynamic entropic risk measure.

Dynamic Capital Allocation and Managerial Compensation

This paper proposes a dynamic theory of capital budgeting and managerial compensation when investment information is decentralized and the agent can obtain private benefit from inefficiently deploying capital. The firm may forgo positive NPV projects, distort pay-performance sensitivity downward, and induce lower investment. The optimal mechanism is to allocate capital and compensate the agent through a budgeting account whose balance reflects the firm's internal financial slack. The agency cost and firm policies all vary monotonically with the account balance. As the firm accumulates more financial slack, fewer projects are forgone, capital is allocated more efficiently to the selected projects, and the agent is provided steeper incentive. The model better explains the documented evidences that firm liquidity is positively correlated with incentive pay, and negatively correlated with investment hurdle rates.

REPRESENTATION OF BSDE-BASED DYNAMIC RISK MEASURES AND DYNAMIC CAPITAL ALLOCATIONS

In this paper, we provide a new representation result for dynamic capital allocations and dynamic convex risk measures that are based on backward stochastic differential equations (BSDEs). We derive this representation from a classical differentiability result for BSDEs and the full allocation property of the Aumann–Shapley allocation. The representation covers BSDE-based dynamic convex and dynamic coherent risk measures. The results are applied to derive a representation for the dynamic entropic risk measure. Our results are also applicable in a specific way to the static case, where we are able to derive a new representation result for static convex risk measures that are Gâteaux-differentiable.

Representation of BSDE-Based Dynamic Risk Measures and Dynamic Capital Allocations

In this short paper we provide a new representation result for dynamic capital allocations and dynamic convex risk measures that are based on backward stochastic differential equations. We derive this representation from a classical differentiability result for backward stochastic differential equations and the full allocation property of the Aumann-Shapley allocation. The representation covers BSDE-based dynamic convex and dynamic coherent risk measures. The results are applied to derive a representation for the dynamic entropic risk measure. Our result are also applicable in a specific way to the static case, where we are able to derive a new representation result for static convex risk measures that are Gateaux-differentiable.

Optimal Design of Internal Capital Markets

I study optimal design of a dynamic capital allocation process in an organization in which the division manager with empire-building preferences privately observes the arrival and properties of investment projects, and headquarters can audit projects at a cost. Under certain conditions, a budgeting mechanism with threshold separation of financing is optimal. Headquarters: (i) allocate a spending account to the manager and replenish it over time; (ii) set a threshold, such that projects below it are financed from the account, while projects above are financed fully by headquarters upon an audit. Further analysis studies when co-financing of projects is optimal and how the size of the account depends on past performance of projects.

Dynamic Capital Allocation With Distortion Risk Measures

Tsanakas and Barnett (2002) employed concepts from cooperative game theory (Aumann and Shapley, 1974) for the allocation of risk capital to portfolios of pooled liabilities, when distortion risk measures (Wang et al., 1997) are used. In this paper we generalise previously obtained results in three directions. Firstly, we allow for the presence of non-linear portfolios. Secondly, based on the concept of correlation order (Dhaene and Goovaerts, 1996) we proceed with discussing the links between dependence structures, capital allocation and pricing, as well as dropping a restrictive assumption on the continuity of probability distributions. Finally, we generalise the capital allocation methodology to a dynamic setting and conclude with a numerical example.

Dynamic capital allocation: exploiting persistent patterns in currency performance

Dynamic capital allocation: exploiting persistent patterns in currency performance

Dynamic capital allocation with distortion risk measures

Tsanakas and Barnett [Insurance: Mathematics and Economics 33 (2003) 239] employed concepts from cooperative game theory [Aumann and Shapley, Values of Non-Atomic Games. Princeton University Press, Princeton] for the allocation of risk capital to portfolios of pooled liabilities, when distortion risk measures [Insurance: Mathematics and Economics 21 (2) (1997) 173] are used. In this paper we generalise previously obtained results in three directions. Firstly, we allow for the presence of non-linear portfolios. Secondly, based on the concept of correlation order [ASTIN Bulletin 26 (2) (1996) 201] we proceed with discussing the links between dependence structures, capital allocation and pricing, as well as dropping a restrictive assumption on the continuity of probability distributions. Finally, we generalise the capital allocation methodology to a dynamic setting and conclude with a numerical example.