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A decision-dependent randomness stochastic program for asset–liability management model with a pricing decision

  • S.I.: Recent Developments in Financial Modeling and Risk Management
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Abstract

In this study, we present a stochastic programming asset–liability management model which deals with decision-dependent randomness. The model focuses on a pricing problem and the subsequent asset–liability management problem describing the typical life of a consumer loan. Such problems are frequently tackled by many companies, including multinationals. When doing so, they must consider numerous factors. These factors include the possibility of their customer rejecting the loan, the possibility of the customer defaulting on the loan and the possibility of prepayment. The randomness associated with these factors have a clear relationship with the offered interest rate of the loan which is the company’s decision and thus, induces decision-dependent randomness. Another important factor, which plays a major role for liabilities, is the price of money in the market. This is determined by the market interest rates. We captured their evolution in the form of a scenario tree. In summary, we formulated a non-linear, multi-stage stochastic program with decision-dependent randomness, which spanned the lifetime of a typical consumer loan. Its solution showed us the optimal decisions that the company should make. In addition, we performed a sensitivity analysis demonstrating the results of the model for various parameter settings that described different types of customers. Finally, we discuss the losses caused if companies do not act in the optimal way.

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Notes

  1. Note that usually the principal \(N_{0}\) is set by the company. However, as it enters the model only as a scale parameter (multiplier of the objective function), we treat it as fixed—for example, determined by a risk management unit of the company.

  2. We should note that we considered four different ratings of a customer, from the best rating (1) up to the worst rating (4). A few customers with a rating worse than 4 on the usual scale \(1 - 8\) were assigned rating 4 for this analysis.

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Authors and Affiliations

  1. Department of Probability and Mathematical Statistics, Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská 83, 186 75, Prague, Czech Republic

    Miloš Kopa & Tomáš Rusý

Authors
  1. Miloš Kopa
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  2. Tomáš Rusý
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Correspondence to Miloš Kopa.

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The work was supported by the Grants GAUK No. 258318 and GAČR No. 18-05631S. Their support is greatly acknowledged.

Appendix

Appendix

In the figures and tables below, we present the results of the sensitivity analysis for other midrates as discussed in Sect. 3.2. First, we show optimal solutions of the program for each rating and interest rate sensitivity of the customer. Then, we present the tables with losses associated with \(\pm 1\%\) difference from the optimal interest rate, the objective value contour plot for the program with a customer of rating 2 and a fixed initial interest rate decision (Figs. 7, 8, 9, 10, 11, 12, 13, 14 ). Objective function surfaces can be seen in the interactive mode available at https://plot.ly/~rusy/.

Fig. 7
figure 7

Sensitivity analysis results for midrate 0.12

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Fig. 8
figure 8

Contour plot constructed from objective values of the program when fixing offered interest rate r for different values of interest rate sensitivity \(b_{1}\)

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Fig. 9
figure 9

Sensitivity analysis results for midrate 0.14

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Fig. 10
figure 10

Contour plot constructed from objective values of the program when fixing offered interest rate r for different values of interest rate sensitivity \(b_{1}\)

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Fig. 11
figure 11

Sensitivity analysis results for midrate 0.16

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Fig. 12
figure 12

Contour plot constructed from objective values of the program when fixing offered interest rate r for different values of interest rate sensitivity \(b_{1}\)

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Fig. 13
figure 13

Sensitivity analysis results for midrate 0.18

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Fig. 14
figure 14

Contour plot constructed from objective values of the program when fixing offered interest rate r for different values of interest rate sensitivity \(b_{1}\)

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Kopa, M., Rusý, T. A decision-dependent randomness stochastic program for asset–liability management model with a pricing decision. Ann Oper Res 299, 241–271 (2021). https://doi.org/10.1007/s10479-020-03583-y

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