Portfolio Optimization:Risk And Capital Requirements

The Global Financial Crisis(GFC)of 2008 reiterated the need and importance for risk mon-itoring and capital adequacy in the banking sector as irrecoverable debt or even bankruptcy have a ripple effect on the overall economy.Now more than ever,matters of lending and the relevance of maintaining adequate capital to absorb adverse market conditions have become imperative to bank managers and financial regulators.Bank interconnectedness and the risk of spillover ef-fects on the real economy fuels the need for banks to manage risk and maintain sufficient capital level necessary to avoid insolvency.Therefore,employing the variance-safety-first risk mea-sure as a way of controlling symmetric risk while providing a safety net against extreme losses is reasonable and worth investigating.To achieve a more accurate estimation of the probability of negative returns,Roy's safety-first principle is modified by determining semi-deviation of the random return from Value-at-Risk(VaR).Using Bienayme-Tchebycheff's inequality,an estima-tion of the probability of downside risk is derived and the estimation is reconstructed to involve the coherent Conditional Value-at-Risk(CVaR).Capital requirements form the building block for financial regulation because capital acts as a cushion to absorb unexpected losses which threaten solvency and affect the real economy.Policy makers have recommended changes to already existing coordinated regulation to avert the probability of occurrence of GFC in future.These changes include a significant increment in Capital to Risk(Weighted)Assets Ratio(CRAR),which is dependent on both the capital re-quired and the capital resources to meet the requirements.As a step in that direction,in the pro-posed consolidated risk measure(variance and the modified Roy's safety-first principle)frame-work,stochastic and distributionally robust portfolio optimization models are presented.By the constraint of CRAR,the portfolio optimization models guarantee a bank,with loans which are supposed to be affinely dependent on uncertainties,treasury bills,fixed assets,and non-interest earning assets,to cope with capital requirements of Basel III with a probability of 95%,thereby not only controlling credit risk but also capital risk.Through auxiliary functions,the chance constraint is redefined to include one uncertain parameter and a mathematical formulation for CRAR chance constraint is derived.It was also showed via theoretical proof that,for the case with a specified probability distribution of loan data and the case where the probability distri-bution of loan data is not completely specified but with given moment information,the CRAR probability constraints can be converted explicitly into deterministic convex second-order cone counterparts and closed-form formulations are obtained.To model the dynamic behavior of loans,a modified CreditMetrics approach to compute the forward loan value and credit risk of unit capital in a multi-period risk horizon was investi-gated,in which all possible credit migration paths in the multi-period risk horizon for the term loans were considered.Pseudocodes for the modified CreditMetrics approach was presented and implemented by MATLAB.This research takes advantage of investor needs and objectives of maximizing returns,mini-mizing risk,providing safety-net against extreme losses among others post-GFC and the trade-off thereof to develop portfolio optimization models.Therefore,series of models and numerical ex-periments implemented by MATLAB have been considered within different contexts to meet the objectives of this study.The mean-variance-modified safety-first model is subjected to norm regularization to seek near-optimal stable and sparse portfolios.For the purpose of providing a numerical illustration for testing the proposed consolidated risk measure,this research initially examines the financial market made up of stocks and compares the results with other selected alternative models in the literature and a benchmark,S&P 500 index.The numerical results ob-tained indicate that the mean-variance modified safety-first model is favorable since it has better out-of-sample performance metrics.The main results obtained in this study suggest that,in the proposed consolidated risk mea-sure of variance and modified safety-first framework,the proposed stochastic and distribution-ally robust CRAR chance-constrained optimization models,guarantee banks of meeting capital requirements of Basel ? with a likelihood of 95%irrespective of changes in the future market value of assets.To administer valuable insights from a financial outlook,this study pursues the scope of analyzing the CRAR chance constrained-optimization models under the worst-case sce-nario.Even under the worst case scenario i.e.when the loans default,the proposed capital to risk asset ratio chance-constrained optimization models meet the minimum total requirements of Basel ?.