Combining a Matheuristic with Simulation for Risk Management of Stochastic Assets and Liabilities

Specially in the case of scenarios under uncertainty, the efficient management of risk 1 when matching assets and liabilities is a relevant issue for most insurance companies. This paper 2 considers such a scenario, where different assets can be aggregated to better match a liability (or the 3 other way around), and the goal is to find the asset-liability assignments that maximises the overall 4 benefit over a time horizon. To solve this stochastic optimisation problem, a simulation-optimisation 5 methodology is proposed. We use integer programming to generate efficient asset-to-liability 6 assignments, and Monte-Carlo simulation is employed to estimate the risk of failing to pay due 7 liabilities. The simulation results allow us to set a safety margin parameter for the integer program, 8 which encourage the generation of solutions satisfying a minimum reliability threshold. A series 9 of computational experiments contribute to illustrate the proposed methodology and its utility in 10 practical risk management. 11

is to find the most efficient (minimum cost) combination of assets that meets certain requirements: 48 they must generate sufficient income to cover the obligations of the insurer with a high probability. In 49 a recent work, Bayliss et al. (2020) considered a simplified ALM problem, based on the net present 50 value (NPV) concept, in which only one-to-one asset-liability assignment were allowed. Notice that, 51 since we are comparing monetary values of assets that belong to different time periods, it makes sense 52 to consider the NPV associated with each asset in order to make a fairer comparison of assets. Our 53 work goes a step further and allows many-to-many, one-to-many, and many-to-one asset-liability 54 assignments as well. Such an approach increases the efficiency with which liabilities can be covered. 55 This also allows us to address ALM problems regardless of the number of assets and liabilities, 56 as well as their sizes. For addressing large scale instances which could not be solved using exact 57 integer programming techniques, previous approaches were based on the use of greedy heuristics that 58 prioritised larger liabilities over smaller ones. This work, however, proposes an improved approach 59 based on sorting liabilities in ascending due date order, since liabilities with earlier due dates have 60 fewer assets combinations that can be assigned to them. Additionally, assets with earlier maturity dates 61 have higher NPVs, which is what is to be minimised. The main methodological contribution of our 62 approach lies in the introduction of a matheuristic algorithm, which integrates integer programming 63 and Monte-Carlo simulation. In particular, an integer program is solved recursively to generate feasible 64 and efficient asset-liability assignments for a deterministic scenario (where we assume average values for ALM, analysing the performance of these models when they are applied to pension funds. institutions that grant loans to general customers. In this paper, stochastic multi-stage scenarios are 140 considered and the behaviour of the consumer are modelled. This behaviour impacts on the decisions 141 the credit institution has to take and how it has to allocate its assets.

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When the conditions set out in a contract are met, insurers pay the insured. If they do not have 144 sufficient available funds, they are subject to monetary fines issued by monetary authorities and, most 145 likely, to lost customers. In order to ensure the insurers can meet their liabilities, they perform a process 146 of matching assets to liabilities. Assigning assets to liabilities in an efficient manner is critical to the 147 success of an insurance firm, since assigned (or frozen) assets cannot be used for any other purpose.

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Assets can only be assigned to liabilities if their maturity date precedes the due date of the liability.

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The value of the assets assigned to liabilities must equal or exceed the liability values. At the same 150 time, asset maturity values and liability payment values are uncertain, thereby introducing a risk that 151 liabilities cannot be met, even when the expected values imply that they could be met on the average.

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An asset-liability assignment is the terminology used in this work to refer to a group of assets used 153 to cover a group of liabilities. A feasible solution to the net present value asset-liability management 154 (NPV-ALM) problem consists of a set of asset-liability assignments such that: (i) all liabilities are 155 covered; and (ii) no individual assets or liabilities are part of more than one asset-liability assignment.

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Furthermore, a solution is also required to be robust under uncertain asset and liability values.

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Specifically, a solution must meet a minimum reliability level, where reliability is defined as the 158 probability that all liabilities can be paid successfully using their assigned assets. Figure 1 illustrates a 159 single asset-liability assignment consisting of three assets and two liabilities. Notice that, under the 160 expected values for assets and liabilities (dashed lines), the liabilities can be met. However, due to 161 uncertain asset maturity values and liability payment values, there is a risk that the assets fail to cover 162 the liabilities in the assets-liability assignments. If f i is the probability that asset-liability assignment i 163 fails to cover its liabilities, then the reliability of a set of asset-liability assignment (I) covering all of 164 our liabilities is computed as r = ∏ i∈I (1 − f i ). Following Faulin et al. (2008), we employ Monte-Carlo 165 simulation to estimate failure probabilities associated with candidate asset-liability assignments.

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In this work, we propose a matheuristic algorithm for solving the NPV-ALM problem. A 167 matheuristic integrates mathematical programming techniques with heuristics in order to develop an 168 algorithm that benefits from exact optimisation as well as from fast and efficient heuristic techniques.

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For the case of the NPV-ALM problem, an integer program (Section 3.2) is used to calculate a set 170 of feasible asset-liability assignment decisions that cover the liabilities. The solution is tested in a 171 simulation to measure its reliability, and the result is employed to tune a safety margin parameter 172 of the integer program. The safety margin parameter controls the minimum ratio between the asset 173 values and the liability values of a generated asset-liability assignment. The process continues until a 174 specified number of iterations have been completed. Section 3.1 formulates the NPV-ALM problem. Probability Probability = 1 + 2 + 3 = 1 + 2 = ൫ሺ − ሻ < 0൯ The uncertain value of asset a at maturitỹ v l : The uncertain value of liability l on its due date Decision variables y ga : Binary variable indicating whether asset a is selected as part of asset-liability assignment g z gl : Binary variable indicating whether liability l is selected as part of asset-liability assignment g w a : Binary variable indicating whether asset a is selected as part of a generated asset-liability assignment x l : Binary variable indicating whether liability l is selected as part of a generated asset-liability assignment Input parameters v a : The expected maturity value of asset a v l : The expected value of liability l on its due date t a : The maturity maturity date of asset a t l : The due date of liability l d : Discount factor used to calculate the net present value of an asset r min : Minimum reliability level m : Safety parameter decrease factor h : Safety parameter increase factor Other parameters f g : Failure probability of asset-liability assignment g N g : Asset-liability assignment g npv g : Net present value associated with Asset-liability assignment g

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The objective (1) is to minimise the NPV of the assets committed to covering liabilities. In this 179 context, y ga is a binary decision variable indicating whether asset a is an element of asset-liability 180 assignment g. Similarly, z gl is a binary decision variable indicating whether liability l is an element of 181 asset-liability assignment g. Each asset a ∈ A can only be part of at most one asset-liability assignment, 182 as specified by Constraint (2). Each liability l ∈ L can only be part of one selected asset-liability 183 assignment, as specified by Constraint (3). As a result of Constraints (2) and (3), the maximum number 184 of asset-liability assignments is |G| = min (|A|, |L|). A feasible asset-liability assignment requires that 185 each of the selected assets matures before all of the selected liabilities in the asset-liability assignment.
186 Constraint (4) introduces a continuous variable φ g representing the latest maturity date of an asset 187 in asset-liability assignment g. Constraint (5) introduces a continuous variable σ g representing the 188 earliest due date of a liability in asset-liability assignment g. Here, H is a large number which ensures 189 the feasibility of Constraint (5) in asset-liability assignments that the liability l is not part of. Then,190 Constraint (6) enforces the time constraints for each asset-liability assignment. Constraint 7 requires 191 that the sum of the asset values exceeds the value of the covered liabilities by a factor S in each asset-liability assignment g, thus ensuring that our liabilities are covered. Also, S is a multiplicative 193 safety margin parameter for ensuring that the asset values are able to cover the liabilities under 194 uncertain asset returns and liability values. Constraints (8) and (9) define the binary decision variables.
3.2. An integer programming model for generating feasible asset-liability assignments asset-liability assignment is generated using the integer program, the selected assets are removed from 205 V and the selected liabilities are removed from U. The integer program is solved repeatedly until 206 the set U is empty. The input k is a randomly selected uncovered liability that must be covered by 207 the next asset-liability assignment generated. This provides a mechanism for randomising the sets 208 of asset-liability assignments generated. The i th asset-liability assignment generated is denoted as N i .

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It contains the set of selected assets and liabilities. The efficiency of an asset-liability assignment is 210 measured by the value of the liabilities covered minus the value of the assets used, which encourages 211 asset-liability assignments to cover as many liabilities as possible with the fewest assets possible.

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The net present value of the assigned assets is then subtracted, which captures our overall objective.

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Higher values of this efficiency measure correspond to more efficient asset-liability assignments. This 214 efficiency objective function is expressed by Objective (10). In this expression, x l is a binary variable

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A feasible asset-liability assignment requires that each of the selected assets matures before the selected 219 liabilities. Constraint (11) expresses this, where t m is the asset maturity date or liability due date of an 220 asset or liability m ∈ V ∪ U. Also, H is a large number which is used to ensure that Constraint (11) 221 remains feasible in cases where liabilities are not selected. Optionally, Constraint (11) can be replaced 222 by a constraint using the same form used in Constraints (4)- (6). 228 Constraint (13) states that the randomly selected uncovered liability, k, must be included in the next 229 asset-liability assignment generated.

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This section describes our matheuristic algorithm, which combines integer programming and Section 3.2) to generate a set of asset-liability assignments that cover the liabilities. This process is 237 iterative, i.e., each iteration generates one new asset-liability assignment from the remaining unused 238 assets and uncovered liabilities. In order to increase the diversity of these solutions, a random factor 239 is introduced: we randomly select one of the remaining liabilities and add a constraint which forces 240 this liability to be part of the next asset-liability assignment. The simulation phase is used to measure 241 the reliability of the generated solution. Monte-Carlo simulation is used estimate the failure probability 242 associated with each asset-liability assignment. This is the probability that the sum of the maturity 243 values of the assets, in an asset-liability assignment, is less than the corresponding sum of the liabilities.

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If the solution is sufficiently reliable, a best solution check is performed to see if the solution has the 245 lowest associated NPV of any reliable solution found. The reliability result is also used to update the 246 safety margin parameter of the integer program. The procedure followed is given in Algorithm 1.

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The proposed heuristic has been implemented as a Python application running on a CPU with 3.60 Algorithm 1: AssetLiabilityAssignmentGeneration (A, L, r min , β, m, h, runs) Data: A set of available assets, L set of liabilities, maxIterations, r min the minimum reliability level, β geometric distribution parameter, m safety margin decrease factor, h safety margin increase factor, runs the number of Monte-Carlo simulation runs used to estimate asset-liability assignment failure probabilities 1 iteration = 1, the number of asset-liability assignments generated so far.; 2 bestSolution ← ∅; 3 bestNPV = ∞; 4 //Initialise the safety margin parameter S = 1; 5 S = 1; 6 while iteration ≤ maxIterations do 7 //Reset the set of unassigned assets V and uncovered liabilities U; plus two new instances that could not be solved with the methodology presented in the former paper.
251 Table 1 provides the details on the number of assets and liabilities for each instance, discount rate, and 252 value modifier (if any was employed). Assets and liabilities have been distributed over time using 253 a random uniform probability distribution from 0 to 100 and from 50 to 150, respectively. Similarly, 254 values for assets and liabilities have been randomly generated using a uniform probability distribution 255 from 0 to 1 and from 0 to 0.5, respectively. Asset values from instances 4 and 5 have been modified 256 to simulate scenarios where its value varies over time, i.e.: given an asset a ∈ A with a value v a at 257 time t a , a new value v a is computed v a = v a f (t a , T), with T = max{t a : a ∈ A} and f the asset value 258 modifier function. Likewise, instances 6 and 7 consider scenarios with liability values varying over 259 time: given a liability l ∈ L with a value v l at time t l , a new value v l is computed v l = v l g(t l , T), with 260 T = max{t l : l ∈ L} and g the liability value modifier function. Instance 10 simulates a scenario with 261 small assets and large liabilities, which encourages the use of multiple assets to cover a liability, while 262 instance 11 considers a scenario with a few large assets and several small liabilities, to force the use of 263 a single asset to cover multiple liabilities. with the geometric probability distribution that drives the liability selection and the relative mixed

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Each instance in Table 1 has been solved using the integer programming algorithm presented 272 in Algorithm 1, with a limit of 100 iterations. A time-limit of 300 seconds has also been imposed to 273 terminate the algorithm after a solution has been generated if the aforementioned time-limit has been for each asset-liability assignment generated in the Monte-Carlo simulation.
279 Table 2 provides the experimental results, compared with the results obtained in Bayliss et al.

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(2020). The first column contains the instance number (same as in Table 1). The second column (Cplex)

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As it can be seen in Table 2, the stand-alone matheuristic is providing reasonably good solutions 289 when compared with the optimal ones given by Cplex for the deterministic scenario. Actually, Cplex 290 is not able to solve all instances since it gets an "out of memory" (OoM) error for instance 3 (which 291 justifies the need of using matheuristics even for the deterministic case). Also, notice that the cost of the 292 assets-to-liabilities mapping is quite different in the deterministic scenario (Det.) and in the stochastic 293 one (Stoch.). In other words, the deterministic scenario represents an 'ideal' (but not realistic) situation 294 that provides a lower-bound to the real NPV cost under uncertainty conditions. Probably, the most 295 interesting comparison in this table is between columns BR and Stoch. As one can see, the proposed 296 matheuristic-simulation algorithm is usually able to outperform the previous simulation-optimisation 297 approach proposed in Bayliss et al. (2020). This is mainly due to the fact that the methodology 298 proposed in this paper does not require to assume a one-to-one mapping between assets and liabilities, 299 thus allowing for an increasing number of mapping combinations. The main benefit of using the 300 matheuristic-simulation algorithm is that it treats reliability as a hard constraint, an issue which is 301 very important in the context of meeting liabilities. However, since the matheuristic is a more complex 302 algorithm than BR, the 300 second time limit meant that there was not enough time for it to find 303 solutions that met the 95% reliability constraint exactly, allowing it to achieve a low NPV. Notice 304 that the gap between the NPVs of BR and the matheuristic are largest when the matheuristic return 305 very reliable solution, while BR returns solution with low reliability. Figure 4 highlights the large 306 average reliability gain attained from using the matheuristic, at the expense of a slightly higher NPV 307 on average.  problem. It also considers the possibility of aggregating different assets, or different liabilities, before 320 completing the assignment mapping, i.e.: several assets can be aggregated to cover each liability, and 321 multiple liabilities can be covered by a single asset. To the best of our knowledge, it is the first time 322 that this many-to-many assignment procedure is considered in the literature on asset and liability 323 management.

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The results show that the best deterministic mapping of assets to liabilities is far from being 325 an optimal solution when uncertainty is present. Hence, simulation-optimisation methods become 326 necessary to generate high-quality solutions whenever some components of the asset and liability 327 management problem need to be modelled as random variables instead of deterministic values. In 328 addition, the numerical experiments show how, by allowing many-to-many assignments between 329 assets and liabilities, our combined matheuristic-simulation algorithm is able to outperform other