Optimizing Asset and Capital Adequacy Management in Banking

This paper considers the application of stochastic optimization theory to asset and capital adequacy management in banking. Our study is motivated by new banking regulation that emphasizes risk minimization practices associated with assets and regulatory capital. Our analysis depends on the dynamics of the capital adequacy ratio (CAR), which we compute in a stochastic setting, by dividing regulatory bank capital (RBC) by risk weighted assets (RWAs). Furthermore, we demonstrate how the CAR can be optimized in terms of bank equity allocation and the rate at which additional debt and equity is raised. In either case, the dynamic programming algorithm for stochastic optimization is employed to verify the results. Also, we provide an illustration of aspects of bank management practice in relation to this regulation. Finally, we make a few concluding remarks and discuss possibilities for further research. The research was supported by a generous grant from the National Research Foundation of South Africa under GUN 2069031.     

Continuous‐time stochastic modelling of capital adequacy ratios for banks

Regulation related to capital requirements is an important issue in the banking sector. In this regard, one of the indices used to measure how susceptible a bank is to failure, is the capital adequacy ratio (CAR). We consider two types of such ratios, viz. non‐risk‐based (NRBCARs) and risk‐based (RBCARs) CARs. According to the US Federal Deposit Insurance Corporation (FDIC), we can further categorize NRBCARs into leverage and equity capital ratios and RBCARs into Basel II and Tier 1 ratios. In general, these indices are calculated by dividing a measure of bank capital by an indicator of the level of bank risk. Our primary objective is to construct continuous‐time stochastic models for the dynamics of each of the aforementioned ratios with the main achievement being the modelling of the Basel II capital adequacy ratio (Basel II CAR). This ratio is obtained by dividing the bank's eligible regulatory capital (ERC) by its risk‐weighted assets (RWAs) from credit, market and operational risk. Mainly, our discussions conform to the qualitative and quantitative standards prescribed by the Basel II Capital Accord. Also, we find that our models are consistent with data from FDIC‐insured institutions. Finally, we demonstrate how our main results may be applied in the banking sector. Copyright © 2005 John Wiley & Sons, Ltd.     

An optimal investment strategy in bank management

This paper considers the application of stochastic optimization theory to asset and capital adequacy management in banking. The Basel II Capital Accord lays down regulations to control bank behaviour, and relies on regulatory ratios such as the capital adequacy ratio (CAR). In an attempt to address the problem of compliance to minimum CAR and under assumptions about retained earnings, loan‐loss reserves, the market and shareholder‐bank owner relationships, we construct a continuous‐time model of the Basel II CAR which is computed from the total risk‐weighted assets (TRWAs) and bank capital in a stochastic setting. In particular, we derive an optimal equity allocation strategy for the bank and monitor the performance of the Basel II CAR under the allocation strategy. Copyright © 2011 John Wiley & Sons, Ltd.     

A generic framework for stochastic Loss-Given-Default

In this document a method is discussed to incorporate stochastic Loss-Given-Default (LGD) in factor models, i.e. structural models for credit risk. The general idea exhibited in this text is to introduce a common dependence of the LGD and the probability of default (PD) on a latent variable, representing the systemic risk. Though our theory can be applied to any arbitrary firm-value model and any underlying distribution for the LGD, provided its support is a compact subset of [0,1], special attention is given to the extension of the well-known cases of the Gaussian copula framework and the shifted Gamma one-factor model (a particular case of the generic one-factor Lévy model), and the LGD is modeled by a Beta distribution, in accordance with rating agency models and the Credit Metrics model.In order to introduce stochastic LGD, a monotonically decreasing relation is derived between the loss rate L, i.e. the loss as a percentage of the total exposure, and the standardized log-return R of the obligor’s asset value, which is assumed to be a function of one or more systematic and idiosyncratic risk factors. The property that the relation is decreasing guarantees that the LGD is negatively correlated to R and hence positively correlated to the default rate. From this relation, expressions are then derived for the cumulative distribution function (CDF) and the expected value of the loss rate and the LGD, conditionally on a realization of the systematic risk factor(s). It is important to remark that all our results are derived under the large homogeneous portfolio (LHP) assumption and that they are fully consistent with the IRB approach outlined by the Basel II Capital Accord.We will demonstrate the impact of incorporating stochastic LGD and using models based on skew and fat-tailed distributions in determining adequate capital requirements. Furthermore, we also skim the potential application of the proposed framework in a credit risk environment. It will turn out that both building blocks, i.e. stochastic LGD and fat-tailed distributions, separately, increase the projected loss and thus the required capital charge. Hence, the aggregation of a model based on a fat-tailed underlying distribution that accounts for stochastic LGD will lead to sound capital requirements.     

基于流动性风险约束的我国商业银行资产负债随机规划模型

通过以资产负债管理合理匹配银行资产、负债,可以防范银行流动性风险.为此,建立了一个带有简单补偿的两阶段多期随机规划,在满足相关政策、法规约束和流动性风险V aR随机机会约束条件下,以银行的盈利最大化为目标,对银行主要资产、负债进行动态的优化匹配.     

Multi-period defined contribution pension funds investment management with regime-switching and mortality risk

Using mean–variance criterion, we investigate a multi-period defined contribution pension fund investment problem in a Markovian regime-switching market. Both stochastic wage income and mortality risk are incorporated in our model. In a regime-switching market, the market mode changes among a finite number of regimes, and the market state process is modeled by a Markov chain. The key parameters, such as the bank interest rate, or expected returns and covariance matrix of stocks, will change according to the market state. By virtue of Lagrange duality technique, dynamic programming approach and matrix representation method, we derive expressions of efficient investment strategy and its efficient frontier in closed-form. Also, we study some special cases of our model. Finally, a numerical example based on real data from the American market sheds light on our theoretical results.     

The optimal operational risk capital requirement by applying the advanced measurement approach

The purpose of this paper is to construct a risk quantification model to achieve the accurate operational risk management and gain the satisfying estimation and control of future possible extreme losses by using capital charges to assess operational risk. The paper takes a case bank as the research object and compares the differences under various circumstances engaging the Basic Indicator Approach, the Standardized Approach, and the Advanced Measurement Approach for the operational risk capital requirement of a bank. The results indicate that it is more appropriate to adopt the Advanced Measurement Approach to estimate the operational risk capital requirement; this way can help a bank enjoy a much lessened capital charge required and subsequently its available capital increases. Hence, it allows a bank to have sufficient funds in operations and reduce the burden of capital costs. Therefore, it will bring the positive benefits to the whole banking industry when enforcing the Advanced Measurement Approach.     

A general asset-liability management model for the efficient simulation of portfolios of life insurance policies

New regulations and a stronger competition have increased the importance of stochastic asset-liability management (ALM) models for insurance companies in recent years. In this paper, we propose a discrete time ALM model for the simulation of simplified balance sheets of life insurance products. The model incorporates the most important life insurance product characteristics, the surrender of contracts, a reserve-dependent bonus declaration, a dynamic asset allocation and a two-factor stochastic capital market. All terms arising in the model can be calculated recursively which allows an easy implementation and efficient simulation. Furthermore, the model is designed to have a modular organization which permits straightforward modifications and extensions to handle specific requirements. In a sensitivity analysis for sample portfolios and parameters, we investigate the impact of the most important product and management parameters on the risk exposure of the insurance company and show that the model captures the main behaviour patterns of the balance sheet development of life insurance products.     

存在借贷资金的随机资金预算问题研究

讨论了企业运用自有资金及银行贷款进行投资时的资金预算问题,与以往的研究不同,本文假设投资支出、年投资收益以及银行贷款都为随机变量,而且,文章的研究并不要求待选的投资项目具有相同的投资期或具有相同的寿命周期.给出了随机环境下净现值收益的期望值模型及期望值目标规划模型,并设计了基于随机模拟的遗传算法,给出了模型的一般解决方法,此外,还提供了两个数值例子,用以说明建模思想,并例证算法的有效性.     

中国银行系统的宏观审慎监管研究

全球金融危机爆发后,对银行系统实行审慎监管已成为国内外学者及相关监管机构的共识。但目前银行系统的监管研究多为微观审慎监管,宏观审慎监管研究缺乏,尤其是对中国银行网络系统进行动态建模并进行宏观审慎监管的定量研究未见。本文首先利用中国2008至2015年16家上市银行的实际数据构建动态的中国银行网络系统模型,然后使用Component VaR、Incremental VaR、Shapley value EL以及ΔCoVaR四种风险分配机制研究中国银行网络系统的宏观审慎监管方法。研究表明:对中国银行网络系统进行宏观审慎监管能够有效提升其稳定性,并且四种机制相比之下,ΔCoVaR的监管效果最为显著,而Incremental VaR则相对较差。此外,通过宏观审慎资本与银行指标之间的相关性分析,发现Incremental VaR、Shapley value EL以及Component VaR机制下的宏观审慎资本与银行的总资产具有一定的相关性,此时宏观审慎资本可以根据银行的总资产来设置;而ΔCoVaR机制下则不相关,因此宏观审慎资本可以依据各银行的系统性风险贡献大小来设置。     

Portfolio choice and optimal hedging with general risk functions: A simplex-like algorithm

The minimization of general risk functions is becoming more and more important in portfolio choice theory and optimal hedging. There are two major reasons. Firstly, heavy tails and the lack of symmetry in the returns of many assets provokes that the classical optimization of the standard deviation may lead to dominated strategies, from the point of view of the second order stochastic dominance. Secondly, but not less important, many institutional investors must respect legal capital requirements, which may be more easily studied if one deals with a risk measure related to capital losses.     

Integrated bank risk modeling: A bottom-up statistical framework

In the last few years, according to the evolution of financial markets and the enforcement of international supervisory requirements, an increasing interest has been devoted to risk integration. The original focus on individual risk estimation has been replaced by the growing prominence of top-down and bottom-up risk integration perspectives. Following this latter way, we bring together different approaches developed in the recent literature elaborating a general model to assess banking solvency in both the long-run (economic capital) as well as in the short period (liquidity mismatching). We consider banking capability to face credit, interest rate and liquidity risks associated to macro-economic shocks affecting both assets and liabilities. Following the perspective of commercial banks, we concentrate on information available in the risk management practice to propose an easy to implement statistical framework. We put in place this framework estimating its scenario generation parameters on Italian macro-economic time series from 1990 to 2009. Once applied to a stylized commercial bank, we compare the results of our approach to regulatory capital requirements. We emphasize the need for policy makers as well as risk managers, to take into account the entire balance sheet structure to assess banking solvency.     

Optimal cash management under uncertainty

We solve an agent’s optimization problem of meeting demands for cash over time with cash deposited in bank or invested in stock. The stock pays dividends and uncertain capital gains, and a commission is incurred in buying and selling of stock. We use a stochastic maximum principle to obtain explicitly the optimal transaction policy.     

Precautionary measures for credit risk management in jump models

Sustaining efficiency and stability by properly controlling the equity to asset ratio is one of the most important and difficult challenges in bank management. Due to unexpected and abrupt decline of asset values, a bank must closely monitor its net worth as well as market conditions, and one of its important concerns is when to raise more capital so as not to violate capital adequacy requirements. In this paper, we model the trade-off between avoiding costs of delay and premature capital raising, and solve the corresponding optimal stopping problem. In order to model defaults in a bank's loan/credit business portfolios, we represent its net worth by Lévy processes, and solve explicitly for the double exponential jump-diffusion process and for a general spectrally negative Lévy process.     

基于动态VaR约束与随机波动率模型的最优投资策略

研究Stein-Stein随机波动率模型下带动态VaR约束的最优投资组合选择问题. 假设投资者的目标是最大化终端财富的期望幂效用,可投资于无风险资产和一种风险资产, 风险资产的价格过程由Stein-Stein随机波动率模型刻画. 同时, 投资者期望能在投资过程中利用动态VaR约束控制所面对的风险.运用Bellman动态规划方法和Lagrange乘子法, 得到了该约束问题最优策略的解析式及特殊情形下最优值函数的解析式; 并通过理论分析和数值算例, 阐述了动态VaR约束与随机波动率对最优投资策略的影响.     

Multi-period mean–variance portfolio selection with regime switching and a stochastic cash flow

This paper investigates a non-self-financing portfolio optimization problem under the framework of multi-period mean–variance with Markov regime switching and a stochastic cash flow. The stochastic cash flow can be explained as capital additions or withdrawals during the investment process. Specially, the cash flow is the surplus process or the risk process of an insurer at each period. The returns of assets and amount of the cash flow all depend on the states of a stochastic market which are assumed to follow a discrete-time Markov chain. We analyze the existence of optimal solutions, and derive the optimal strategy and the efficient frontier in closed-form. Several special cases are discussed and numerical examples are given to demonstrate the effect of cash flow.     

Corporate valuation,capital structure and risk management: A stochastic DCF approach

In this article we discuss a general stochastic framework for designing corporate investment, financing and risk management strategies for financially constrained firms. The strategy entailing the highest benefits for shareholders is considered to be the optimal strategy. This paper focuses on a simulation of present value distributions of the capital positions of a company, explicitly taking into account the risk of fluctuations in future cash flow as well as the risk of insolvency. The present value distribution of equity is used as a central instrument for evaluation of shareholder benefits. Expected present values are also computed. The investment and financing policy of the company pursued at the time of the valuation is reflected in certain global model parameters, which themselves influence the future profit distribution policy of the company. The main parameters are the extent of debt, the annual debt funding requirements, the average earnings power of the company – expressed as an expected annual return on total capital – and the risk of annual earnings – expressed as the standard deviation of the annual return on total capital. An explicit illustration of the volatility risk and default risk seems not only to be a suitable way of illustrating the impact of capital structure on corporate value. Such an depiction may also provide answers to the question of the link between hedging and enterprise value. This paper highlights the fact that investment, finance and hedging strategies should go hand in hand.     

A mixed integer programming model for multistage mean–variance post-tax optimization

In this paper, we introduce a mixed integer stochastic programming approach to mean–variance post-tax portfolio management. This approach takes into account of risk in a multistage setting and allows general withdrawals from original capital. The uncertainty on asset returns is specified as a scenario tree. The risk across scenarios is addressed using the probabilistic approach of classical stochastic programming. The tax rules are used with stochastic linear and mixed integer quadratic programming models to compute an overall tax and return-risk efficient multistage portfolio. The incorporation of the risk term in the model provides robustness and leads to diversification over wrappers and assets within each wrapper. General withdrawals and risk aversion have an impact on the distribution of assets among wrappers. Computational results are presented using a study with different scenario trees in order to show the performance of these models.     

Optimal portfolio choice and stochastic volatility

In this paper we examine the effect of stochastic volatility on optimal portfolio choice in both partial and general equilibrium settings. In a partial equilibrium setting we derive an analog of the classic Samuelson–Merton optimal portfolio result and define volatility‐adjusted risk aversion as the effective risk aversion of an individual investing in an asset with stochastic volatility. We extend prior research which shows that effective risk aversion is greater with stochastic volatility than without for investors without wealth effects by providing further comparative static results on changes in effective risk aversion due to changes in the distribution of volatility. We demonstrate that effective risk aversion is increasing in the constant absolute risk aversion and the variance of the volatility distribution for investors without wealth effects. We further show that for these investors a first‐order stochastic dominant shift in the volatility distribution does not necessarily increase effective risk aversion, whereas a second‐order stochastic dominant shift in the volatility does increase effective risk aversion. Finally, we examine the effect of stochastic volatility on equilibrium asset prices. We derive an explicit capital asset pricing relationship that illustrates how stochastic volatility alters equilibrium asset prices in a setting with multiple risky assets, where returns have a market factor and asset‐specific random components and multiple investor types. Copyright © 2011 John Wiley & Sons, Ltd.