1. Introduction
Undoubtedly, after the Global Financial Crisis (GFC) of 2007–2008, the spotlight on capital requirement heightened. The Global Financial Crisis was caused by “sub-prime” housing loans in the form of mortgage-backed securities. Numerous determinants for the Global Financial Crisis have been proposed, with various weights assigned by researchers [
1]. The Financial Crisis Inquiry Commission stated that the Global Financial Crisis was avertable and its root cause was “widespread failures in financial regulation and supervision...”. A persistent deficit of bank capital and a 25% dip in private investment on average comprise some of the aftermaths of GFC [
2]. In tackling the problems and loopholes in financial regulations unveiled by the GFC, Basel III was proposed. Its main objective is to bolster bank capital requirement by expanding bank liquidity and lessening bank leverage. The changes in Basel III include a meaningful surge in the Capital to Risk (weighted) Assets Ratio (CRAR) [
3].
The utilization of CRAR safeguards depositors and improves the efficiency and stability of financial frameworks. In [
4], a CRAR chance-constrained optimization model was proposed to guarantee that a bank can cope with the capital requirements of Basel III with a probability of 95%, irrespective of the changes in the future market value of assets. The proposed model considered loans having truncated Gaussian distributed returns, which allowed reformulating the chance constraint related to capital requirements in a second-order cone condition. For the purpose of completeness, the CRAR chance constraint is re-introduced:
where
and
.
is the bank’s total liability, and
M is the bank’s total asset amount. For the purpose of this study, Tier 1 capital (core capital) consists of shareholders’ equity and disclosed reserves,
. Tier 2 (supplementary capital) consists of revaluation reserves,
, and general loan loss provisions,
. Denote
as the vector of the annual interest rate of loans and the treasury bill, fixed assets, and non-interest earning assets (riskless).
is the vector of assets with
and
corresponding to loans and riskless assets (treasury bill, fixed assets, and non-interest earning assets), respectively.
constitutes uncertain parameters that can be estimated, and
is a deterministic vector of
. The Basel III total capital requirement ratio is denoted as
;
is the
asset’s weight factor; and
is the safety factor. Denote
as the vector of asset allocation or investment proportion, which is the decision variable.
The work in [
4] assumed that the full and accurate probability distribution of the random vector
is known, given estimations from historical data and information from the literature. However, one might have only partial information about the probability distribution: its moment information. Therefore, replacing an unknown distribution with a particular distribution might lead to an over-optimistic solution, resulting in an unsatisfactory chance constraint under the true or actual distribution of random vector
. The work in [
5] stated that a more difficult challenge that arises is the need to commit to a particular distribution of random vector
given only restricted information about the stochastic parameter. To avoid the difficulty of selecting a proper distribution and uncertainty surrounding it, the work in [
6] explored the distributionally-robust optimization approach. In this approach, after defining a set
of possible probability distributions that are assumed to include the true probability distribution
of random vector
, the optimization problem is reconstructed with respect to the worst case expected function over the selection of the probability distribution in this set. Uncertainty in parameter
is described through uncertainty sets that contain many possible values realized for random vector
. When the uncertainty set is characterized by statistical estimates of the mean and covariance, the work in [
7] provided a sufficient condition to guarantee the satisfaction of the constraint with distribution uncertainty at a specified confidence level.
A natural way to tackle a chance constraint against parameter uncertainty is to use the constraint robustness approach. In particular, the distributionally-robust CRAR chance constraint can be expressed as:
where
denotes the set of all probability distributions that are consistent with the first and second moments of the probability distribution of the random vector,
. Whenever
x satisfies (
2) and
is the true distribution,
x satisfies the chance constraint (
1) under true probability distribution
. The work in [
8] revealed that contrary to the stochastic programming approach, the distributionally robust chance constraint reflects investors’ risk and aversion towards exposure to uncertainty about the probability distribution of the outcomes via consideration of the worst probability distribution within
. Thus, this study aims to close the research gap by employing the distributionally-robust approach as a way to avoid the difficulty of selecting a proper distribution and uncertainty surrounding the random vector in the framework of meeting capital requirements.
Empirical evidence indicates that the failures of several banks during the GFC kindled concern about maintaining the excessive risk-taking behaviour of banks. Thus, employing variance-Roy’s safety-first risk measure as a way of minimizing risk while providing a safety net against extreme losses is reasonable and worth investigating. Therefore, we employed the modified and improved Roy’s safety-first principle investigated by [
9]. It is important to note that appropriately modelling risk and meeting capital requirements among other objectives are important for financial stability and that an economy with an efficient financial market structure develops faster.
This paper aims to extend the proposed single-objective CRAR chance constrained optimization model in [
4] by considering the multi-objective constraint robustness approach in a modified safety-first framework. This paper also considers credit risk and the expected value of the portfolio. In solving the model,
Section 3 introduces steps in constructing a deterministic convex counterpart of robust the probability constraint (
2).
In summary, this paper considers a multi-objective distributionally-robust chance-constrained model for capital adequacy. A deterministic equivalent of the robust chance constraint is developed, and computational results are provided to suggest that the model is effective at generating capital allocation decisions even under the worst case realizations (i.e., default) of the debt instruments considered.
The structure of this study is organized as follows: in the next sections, literature review on optimization under uncertainty is discussed, and problem definition and assumption are presented.
Section 4 provides the model formulation and approach, and the next section presents the development of the model. Numerical examples and computational results of our method are shown in
Section 6. The last section concludes the paper.
2. Literature Review: Optimization under Uncertainty
Dependent on objectives, constraints, and decision variables, the literature on deterministic programming models categorizes problems as linear programming [
10], non-linear programming [
11], and integer programming [
12], among others. However, real-life data are usually not certain, and some methods have been proposed for treating such parameter uncertainty. One conventional approach is sensitivity analysis [
13], which deals with uncertainty after finding the optimal solution.
Other frameworks that explicitly incorporate uncertainty into the computation of the optimal solution are stochastic programming, dynamic programming, and robust optimization [
14]. Although the above-mentioned methods overlap, they have unfolded independently of each other. Stochastic programming incorporates stochastic components into the programming framework. The method represents uncertain data by scenarios via for example, Monte Carlo sampling, and simple average approximation. Dynamic programming deals with stochastic uncertain systems in a multi-stage framework. It is a technique more widely utilized in derivative pricing as it tackles problems with uncertain coefficients over multiple horizons. In recent times, robust optimization method is a widely acceptable approach in tackling uncertainty. Robust optimization models uncertainty by using a certain membership (uncertainty sets that are based on statistical estimates and probabilistic guarantees on the solution) and optimizes the worst possible case of the problem. When the uncertain parameters are known within certain bounds, robust optimization is best suited [
14].
Let us consider a general stochastic programming problem:
where the expectation is taken over
. Here, the objective function
is dependent on decision variable
x and uncertain parameter
. The objective function is well defined, as it is optimized on the average. An important question often asked is what if the uncertainty resides in the constraints [
15]. One approach is to formulate such problems similarly by incorporating penalties for constraint violations [
9]. An alternative approach also employed in this study is to require that the constraints are satisfied for all possible values of the uncertain parameters with a high probability. Contemporary work in robust optimization has resulted in defining and specifying uncertainty sets to guarantee that chance constraints are satisfied with a targeted probability, thus providing a connection between stochastic programming and robust optimization. For the purpose of this study, the theory of chance-constrained models is explored further.
The chance-constrained stochastic optimization method is one of the major approaches to solving optimization problems under uncertainty. It ensures that an individual constraint is satisfied with a target probability. Mainly, it restricts the feasible region so that a solution is obtained at a high probability. Chance-constrained programming was first investigated by [
16] to ensure that the optimal solution satisfied constraints at a certain probability or confidence level. Many research works have now delved into more ways of tackling chance-constrained problems and increasing the efficiencies of such optimization problems.
A general chance-constrained programming problem takes the form:
where
x denotes decision variables,
denotes a set of all feasible solutions,
represents uncertain parameters, and
is a desired safety factor chosen by the modeler. The chance constraint ensures that the constraint
is satisfied with a probability
at least.
Chance-constrained optimization problems are challenging computationally. Even checking the feasibility of a chance constraint is NP-hard, and the feasible region is usually non-convex. It is also difficult to obtain samples to estimate the uncertain parameter’s probability distribution accurately. In practice, assumptions about the probability distribution of the uncertain parameters in a chance-constrained problem need to be made to express the probabilistic constraint (
5) in closed form. It is, however, difficult to obtain an equivalent deterministic constraint for most probability constraints.
A more difficult challenge that arises is the need to commit to a particular distribution of the uncertain parameter
given only restricted information about the stochastic parameters [
5]. To avoid the above difficulties such as a selection of the proper probability distribution of the uncertain parameter, NP-hard feasibility checking, and nonconvexity, approximation methods have been proposed. In general, there exist two kinds of approximation approaches for a chance constraint: the analytical approximation method and the sampling-based method. Given the disadvantages of the sampling-based method such as the use of an emprical distribution of the random samples to model the actual distribution [
17] among others, we pursue the analytical approximation approach. The analytical approximation method formulates the chance constraint into an equivalent deterministic counterpart. Robust optimization presents a way to approximate analytically a chance constraint. This technique requires a mild assumption on the probability distribution of the uncertain parameters and provides a tractable and feasible solution to the chance-constrained problem. Research contributions using the framework of robust counterpart optimization were explored by [
18,
19].
A natural way to tackle a chance constraint against parameter uncertainty, which is itself characterized by an uncertain probability distribution, is to use the Distributionally-Robust Chance-Constrained (DRCC) approach, a variant of distributionally-robust optimization. Distributionally-robust optimization is an approach that bridges the gap between robust optimization and stochastic programming [
20]. In particular, the distributionally-robust chance constraint can be expressed as:
where
represents a set of all probability distributions that is in line with the characteristic properties of the true probability
such as moment information or its support [
21]. Whenever
x satisfies (
7) and
is the true distribution,
x satisfies the chance constraint (
5) under the true probability distribution
.
In the DRCC paradigm, the distribution of
is not exactly known, but rather assumed to belong to a given set
. In other words, Equation (
7) requires that for all probability distributions of
, the chance constraint holds. In the DRCC framework, the work in [
22] investigated safe tractable approximations of chance-constrained affinely-perturbed linear matrix inequalities. The work in [
7] showed that in some cases of a linear chance constraint problem, the worst case moment expression could be analytically expressed. Based on S-lemma, the work in [
23] showed that a distributionally-robust chance constraint is tractable when
is linear in the decision variable
x and piecewise linear or quadratic in the uncertainty parameter
. In this study, to obtain a well-posed optimization problem without assuming full knowledge of the probability measure, in moment-based optimization, a distributionally-robust counterpart to a defined chance constraint of capital requirement is considered to guarantee satisfying bank capital requirements.
5. Model Development
The asset-liability optimization model based on the chance constraint of capital to the risk (weighted) asset requirement is structured and presented in this section. Objective functions consider maximizing the annual interest rate and expected portfolio value, minimizing credit risk, and providing a safety-net against extreme losses.
Given definitions of input parameters and replacing chance Constraint (
1) with its robust constraint counterpart (
2) provide the following distributionally-robust chance-constrained problem:
Based on Theorem 1, the robust chance constraint is equivalent to a deterministic convex counterpart. Therefore, the multi-objective robust chance-constrained optimization model is:
where
,
,
, and
.
Given a portfolio
x for
u number of loans, the probability of the loss function not exceeding an acceptable threshold
is:
For a confidence level
, the VaR of portfolio
x is given by:
The corresponding CVaR is expressed as conditional expectation of the loss of the portfolio exceeding or equal to VaR, i.e., when all random values are continuous, the following is derived:
The work in [
33] proposed an equivalent function for CVaR. They expressed their idea as:
where:
One can find the optimal CVaR by solving the right-hand side of Equation (
18). In order to minimize CVaR over
x, we minimized the auxiliary function with respect to
x and
:
The work in [
33] presented an approximation to the auxiliary function
via the sampling method:
Comparing Equation (
20) to Equation (
18), the problem
can be approximated by replacing
with
in Equation (
19):
To solve this optimization problem, one can replace
with artificial variables
and impose constraints
and
:
For a portfolio of u loans, we assumed is the random vector of the expected returns value of with a probability density function . To determine the mean loss of the portfolio, this study defines the loss function as . Since the loss function is convex, then the auxiliary function is a also a convex function and can be solved using well-known optimization techniques.
Representing each objective function by
, the multi-objective robust chance-constrained optimization problem (
23) can be approximated via the approximation technique (
22) as shown below:
where
,
,
, and
.