Abstract
In this paper, we consider a bank asset allocation problem with uncertain migration risk of credit ratings and capital adequacy ratio (CAR) regulations. In the practical scenarios, the future market values of each risky asset are largely affected by outer complex environments. We only observe the information about their first-moment and marginal second-moment of year-ahead market value of each loan asset. Based on these scenarios, we propose a new distributionally robust optimization model with the chance constraint characterized by uncertain CAR. Following the duality theory in infinite-dimensional optimization problem and the theory of conic linear optimization model, we can reformulate the original problem into a tractable linear deterministic semi-definite programming (SDP) model. By using this tractable linear SDP model, we can provide a robust asset allocation policy to bank managers. Further, we conduct a simulation study to illustrate the application of our method under two different economic conditions, a downward condition and an upward condition. Then a series of sensitivity tests is applied to examine the impacts of various factors, including safety probability, target CAR and recovery rate, on the optimal asset allocations. We also compare the performance of our model and the CVaR model. We demonstrate our model provides an efficient way to deal with the trade-off between expected return and CAR.
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Notes
For brevity, “maximize” and “minimize” are replaced by “max” and “min” in the rest of the paper.
In the simulation studies, for simplicity, we only consider 13 assets. We acknowledge that a real bank asset allocation problem may involve a large number of assets. To avoid large-scale problem, those assets sharing common characteristics can be combined into a particular asset group before applying our model. The large-scale asset allocation problems have been considered in the literature (Sirignano et al. 2016), but it is out of the scope of this study.
It is not necessary for a bank to use the Basel II standardized method to calculate its CAR. There are other methods used in practice to calculate the CAR, such as the internal rating based approach. In this study, we assume the ABC bank adopts the standardized method, and all our conclusions are based on this assumption too.
This statement is valid, especially when CAR is calculated by the standardized approach. Further examination is required in order to obtain similar conclusion if CAR is calculated by other methods such as internal rating based approach.
It is worthwhile to notice that the calculation time of a CVaR problem has been speeded up, see for example, Alexander et al. (2006). In this simulation study we adopt a conventional calculation method for the CVaR problem. The calculation time can be reduced if a more recent calculation method was applied.
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Acknowledgments
This work was supported in part by the NSFC under Grants 71671023, 71931009, 71731003, and 71301017, in part by the Foundation for Innovative Research Groups of NSFC under Grant 71421001, in part by China Postdoctoral Science Foundation of Grant 2016M600207, the development Foundation of teaching and scientific research for teachers of Liberal Arts in Zhejiang University and the Fundamental Research Funds for the Central Universities under Grant DUT19LK50.
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Appendices
Appendix A: Proof of Proposition 1
In order to keep a self-contained and convenient analysis in the context, we copy the left-hand-side item of inequality (17) with the set of probability distribution D (16) as follows.
where
By using the probability theory, we can equivalently transform the optimization problem (17’) under the constraint (16’) into the following optimization problem:
where \( \int_{{R^{n} }} {} \) represents n dimensional integral over real field. For the sake of simplicity, domain of the integral, Rn is dropped in the remainder of this section. I(-∞,0] in (24a) stands for the indicator function whose value equals 1 when the value of y0(x) + y(x)Tξ falls into(-∞,0] otherwise it equals 0. In (24d), \( \left\langle \cdot \right\rangle \) refers to inner product between matrices and is equivalent to the trace of the product of two matrices; and ek is a n × 1 unit vector where only the k-th entry is 1 and the remaining entries are 0.
Define the Lagrangian function of the optimization problem (24a)–(24e) as
Base on the above Lagrangian function, we construct the following optimization problem
where θ,α,β are the lagrangian multipliers (θ is a real number; and α, β are n × 1 vectors).
According to Convex Analysis (Rockafellar 1970), we can easily know that the above problem (25) is equivalent to the problem (24a)–(24e). Indeed, problem (25) implies that constraints (24b)–(24d) must be satisfied. According to the dual theory from Convex Analysis (Rockafellar 1970), the dual problem for the optimization problem (25) can be written as the following form:
Furthermore, the optimization problem (26) can be equivalently converted as
where Diag(β) denotes diagonal matrix with the vector of β being the diagonal elements. The optimization problem (27) can further be transformed to the following optimization problem:
Indeed, the constraint (29) can be equivalently transformed into the following forms:
and
for any dF(ξ) ≥ 0,and hence the minimum of the inner problem of the above problem (27) is zero. Therefore, the objective function of the problem (27) becomes θ + αTμ + βTρ in this case.
Note that the semi-infinite constraint (29) can be easily transformed into the two equivalent semi-infinite constraints as follows:
Further, we can equivalently transform the inequality (30) into:
where“≺0” stands for negative semi-definite, and equivalently transform the inequality (31) into:
The above constraint (32) implies that Diag(β) is a negative semi-definite matrix, hence the left-hand-side of the inequality (33) is a concave quadratic maximization problem. Furthermore, by introducing Lagrange multiplier λ and the dual problem of the maximization problem of the left hand side of the inequality (33), the inequality (33) has the following equivalent form
or
The inequality (34b) implies that there exists some λ ≥ 0 and for arbitrary ξ the following inequality holds
Furthermore, inequality (35) is equivalent to:
for some λ ≥ 0. In (36), “≺0” stands for negative semi-definite. Through the above interpretation, the dual problem (26) or the problem (28)–(29) is equivalent to the following problem:
The remaining job in order to complete the proof of Proposition 1, is to prove that the primal problem (24a)–(24e) or (25) and its dual problem (26) is equivalent. We set φ = I(-∞,0](y0(x) + y(x)Tξ), ψ1(·) = 1, b1 = 1, ψ2(·) = ξ, b2 = μ, ψi(·) = ξ 2i-2 , bi= ρi-2 (i = 3,…,n + 2). We can easily prove that functions ψ2,.., ψn+2 are continuous and φ is upper semicontinuous. Further, we can know that the primal problem is consistent. According to the Corollary 3.1 of Shapiro (2001), we can prove that there is no gaps between the primal optimization problem (24a)–(24e) and it’s dual optimization problem (26). Based on the above analysis, we can prove that the chance constraint (17) with the uncertain set (16) is equivalent to
Further, by using a simple analysis, the inequality (38) can be equivalently rewritten as follows: there exist θ∈R, α∈Rn,β∈Rn and λ ≥ 0 such that θ + αTμ+βTρ ≥ 1 − ε and constraints (37b)–(37c) are satisfied. Thus, the proof of Proposition 1 is completed.
Appendix B: The Proof of Theorem 1
Let θ* = θ/λ, α* = α/λ, β* = β/λ and λ* = 1/λ, (19a)–(19c) in proposition 1 are equivalently transformed to (21b)–(21d) in theorem 1 for the case of λ > 0 and x, θ*, α*,β* and λ*(> 0) become decision variables. In fact, the constraint (34a) or (34b) implies that there exists some λ > 0 such that
otherwise for any λ > 0,
If (39) holds for any λ > 0, then the minimization of the problem described as the left-hand-side of inequality (34a) can be obtained only when λ = 0; thus inequality (34) is equivalent to
i.e.,
and the equivalent form (38) of CAR constraint (17) or (15) becomes
which implies CAR constraint has relation with lagrangian multipliers but no any relation with investment proportions x i.e., there is virtually no restriction on the optimal investment proportions of CAR constraint. Therefore, the constraint (19c) in Proposition 1 for some λ > 0 is required from practical perspective and it is equivalent to the constraint (21d). Besides, the equivalence of (19a) and (21b), (19b) and (21c) is obvious, if let θ* = θ/λ, α* = α/λ, β* = β/λ and λ* = 1/λ. Finally, combining the objective function (14), the transformed CAR constraints (21b)–(21d) and other constraints (20), the asset allocation optimization model is reformulated as tractably linear SDP problem as (21a)–(21g). The proof of Theorem 1 is completed.
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Yan, D., Zhang, X. & Wang, M. A robust bank asset allocation model integrating credit-rating migration risk and capital adequacy ratio regulations. Ann Oper Res 299, 659–710 (2021). https://doi.org/10.1007/s10479-020-03571-2
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DOI: https://doi.org/10.1007/s10479-020-03571-2
Keywords
- Asset allocation
- Credit rating migration
- Capital adequacy ratio
- Distributionally robust optimization
- Chance constraint