مدیریت ریسک پویا در سیاست نرخ بهره یک نمونه کار در تعامل با وام از طریق استراتژی سرمایه در چهارچوب تصادفی گسسته

Lending rate policy via an appropriate investment strategy for an interacted portfolio of loans into discrete stochastic framework is examined in this paper. A bank optimization model with several control variables, stochastic inputs and a smoothness criterion described by a quadratic functional is proposed for managing the task. The state variable of the system corresponds to the accumulated surplus profit or loss can oscillates deliberately absorbing fluctuations in the different parameters involved. The theoretical model is solved using standard linearization and advanced stochastic optimization techniques resulting in analytic formulae for the control variables. These solutions are actually feedback mechanisms of the past accumulated surplus profit or loss of each sub-portfolio of loans. At the end, a numerical application is presented deriving a smooth solution for the development of the controllers.

In recent years, more and more financial institutions have devoted important resources to manage in an appropriate way their lending rate policy for the different portfolios of loan in an effort to maximize profits, and to mitigate the ill effects of allocation inefficiency that may arise from changes in international, national and local economic and business conditions, changes in the trend, volume and severity of past due loans and loans graded as low quality, changes in the experience, ability, and depth of lending management and staff, changes in the credit risk profile of the loan portfolio as a whole etc, see Bank for International Settlements (2006). In a parallel direction, the Federal Reserve Board (2006) claims that ensuring stronger capital levels and good risk management at the banking organizations are critical to the health of the banking and financing system. The respective literature for optimal loan models considering default-risk and asset portfolios is very rich. A brief presentation of the most important papers is provided in the next paragraphs. Sealey (1980) presents a model of the depository financial intermediary under uncertainty, where its distinguishing characteristics are the deposit rate-setting behaviour, resources costs, and non-linear risk preferences. It is obvious that the above parameters involved play a crucial role in determining the optimal loan portfolios and deposit rate decisions. Ho and Saunders's (1981) paper showed that the intermediation margin depends on four important factors: the degree of managerial risk aversion; the size of transactions undertaken by the bank; the bank market structure; and the variance of interest rates. Furthermore, the major implications of Slovin and Sushka's (1983) research work are that the commercial lending rates are primarily a function of open market interest rates and under normal conditions the settings of the loans are dichotomized from conditions in deposit markets. Few years later, Allen (1988) expands Ho and Saunders's (1981) model by showing that the pure interest rate spreads may be reduced when cross-elasticities of demand between bank products are considered. Among Zarruk's (1989), and Zarruk and Madura's (1992) findings are that the changes of bank capital, bank capital requirements and deposits as well, lead to the analogous changes of intermediation margin, and unrelated changes in borrowing and lending margins, respectively. Wong (1996) explores the determinants of optimal bank interest margins by constructing a simple firm — theoretical model under multiple sources of uncertainty and risk aversion. He comes to the conclusion that the bank interest margin is positively related to the bank's market power to the operational costs, to the degree of credit risk, and to the degree of interest-rate risk. Angbazo (1997) empirically confirms that banks with more risky loans and higher interest-rate risk exposure would select loan and deposit rates in order to achieve higher net interest margins. Additionally, Nakamura et al. (2001) propose a stochastic model which evaluates quantitively the expected cost, including the costs for bankruptcy and mortgage collection, and they discuss the adequate lending rate analytically and numerically. Finally, Machauer and Weber (1998) provide sufficient evidence for the relation between the loan terms and the risk that borrowers are willing to take. Analyzing data from five leading German banks, they found that lending rate premiums and lines of credit are related to borrower credit ratings while collateral showed no clear relation. Quite recently, in Edelstein and Urosevic's (2003) research work, the optimal lending rate contracts under the conditions of risky, symmetric information for multi-period (dynamic) models is analyzed. According to their work, the optimal lending rate depends on the volatility, and co-variation among the market interest rate, borrower collateral, and income, as well as the time horizon and the risk preferences of lenders and borrowers. Moreover, Stanhouse and Stock (2004) take into consideration the determination of optimal loan and deposit rates, as well as the phenomenon of loan prepayments and deposit withdrawals. Furthermore, the bank institution managers should be compensated with the risk that borrowers are not always consistent with their repayments. It is obvious that loans are priced according to the involved risks, and the capital profits that the management desires. However the most recent time period a contradictory question is always being raised about whether the financial institutions should provide cheap loans in order to attract more customers for other profitable business or not. Something really interesting has been shown by Fried and Howitt (1980), and Petersen and Rajan (1995). They showed that the welfare is enhanced by smoothing of lending rates in relation to borrower risk and market interest rates. In this paper, the main contribution is to introduce and develop a general stochastic discrete-time, large-scale model for managing the lending rate policy for different sub-portfolios of loans in a way that financial institution managers are seeking for. Thus, the paper connects the lending rate policy of an interacted portfolio of loans with discrete stochastic control theory. Although optimal control theory was developed by engineers to investigate the properties of dynamic systems of difference or differential equations, it has also applied to financial problems. Nowadays with the vast developments in computer science, more and more large-scale macroeconomic models were being developed and widely used for forecasting and policy analysis. Moreover, if we take also into consideration these developments and the important theoretical tools, such as stochastic dynamic programming, the linear quadratic programming, the state feedback controllers etc., the introduction of the control theory into the economic systems began a much more attractive proposition. Tustin (1953) was the first to spot a possible analogy between the industrial and engineering processes and post-war macroeconomic policy-making (see Holly and Hallett, 1989, for further historical details). More recently, in the vast literature of banking, Jobst et al. (2006) develop a modeling paradigm which integrates credit risk and market risk in random dynamical framework and use multistage stochastic programming tools. From this point of view, a method of controlling over time some major variables is introduced buffering any kind of fluctuations, in order to absorb partially or completely the probable unexpectedness in micro- and/or macro-economic conditions, in external factors as competition, legal and regulatory requirements or other worsen random events. Moreover, the financial institution managers desire to keep the profit for the bank close to a specified trajectory. Furthermore, since the bank has a certain total capacity for providing loans, and its customers are not always consistent with their repayments, at each time, a different amount is repaid through the installments. This amount is normally smaller, but at some exceptions may take values greater, as a consequence of borrowers paying with some time delay two, three or more installments to the bank. Moreover, the financial institution has to pay several kinds of operational expenses, to contribute the operational cost, and to give back the rate of return to customers due to bank deposits. Additionally, it has the alternative to earn extra money through an insightful investment strategy for the surplus profit of loans. Thus, the model should be enriched a lot by the introduction of an appropriate balanced and active investment strategy, and further variables are also established for the management of the system. Hence, it is constructed for each sub-portfolio of loans as a comprehensive and convenient model with three control variables and ten inputs random (stochastic) and deterministic variables. These three variables; the portion invested in bonds and in shares, and the supplementary, positive or negative (diminishing the main lending rate) valued lending rate are to be controlled through a smooth path over time. For each one of the control variables an analytic formula is also derived making the problem practically approvable, since the use of these formulae can easily manage the state of the system in directions that practitioners and regulators will. As a matter of fact, the control and the relative modification of these variables are not an easy task, as any solution or action should be acceptable by society in general and by the financial institution managers who are going to meet the challenge. Hence, the smoothness of the path is very important and is determined by a functional, which weights changes in the three variables. The weights are key parameters which reflect the expectations of all participants in the banking system as well as the underlying secular trends. A brief outline of the paper is as follows. Section 2 provides the incentives and the typical modeling features of the problem, and it also presents the linearization technique and concludes to the final difference equation. Section 3 offers a detailed study of the solution of the model in the general case, providing an interesting theoretical example with m sub-portfolios of loans. A sophisticated numerical application is described in Section 4, with some interesting and insightful diagrams. Finally, Section 5 concludes the whole paper.

This paper presents both a theoretical and a practical approach for controlling the lending rate policy into an interacted portfolio of loans and the investment strategy into a discrete-time stochastic environment. The introduction of some controllers which act as a kind of buffer absorbing stochastic or other fluctuations in the parameters of the system improves effectively the overall performance of the system. As regards the theoretical model we design: (a) State vector which corresponds to the evolution of accumulated surplus profit or loss for m sub-portfolio of loans. (b) Interaction among the m loan sub-portfolios. (c) Three control variables for each of the loan sub-portfolio which correspond: • Two of them, the proportions invested in bonds and shares. • The supplementary lending rate. (d) Ten input variables: • Three, for the investment rates of returns for cash, bonds and shares. • The basic lending rate for each sub-portfolio of loans. • The ratio of the total amount of commercial loans which correspond to the installments (or not) paid. • The capital cost (including expenses, operational cost, rate of return to customers due to bank deposits and the desirable profit for the bank). • The percentage of surplus profit or loss transferred from the i loan sub-portfolio to j sub-portfolio and the corresponding cost. • The total amount for the whole portfolio of loans, and • The ratio of the total amount placed to the each loan sub-portfolio. (e) A functional that corresponds to the objective function of the problem. In our model the functional contains three smoothing variables (the same as the controllers) and a desired final accumulated surplus profit or loss, for each loan sub-portfolio. As regards the practical application for the two loan sub-portfolio, it is modeled: (a) The rates of the investment returns, the rate of return to customers due to bank deposits, the ratio to the total amount correspond to the installment (or not) paid and the capital cost assuming stationary (unconditional) autoregressive processes in discrete-time of order 1. (b) Additionally, the basic lending rate, the total amount for the whole portfolio of loans, the total amount placed to each loan sub-portfolio, the capital cost, the transferring percentage and the corresponding transition cost are obtained to be constants, following the respective international political and financial agreements. The theoretical optimal solution for the three control variables is determined via certain feedback mechanisms with respect to the past values of the whole sub-portfolio. These analytic solutions, although complicated, with a lot of parameters involved, provide useful insight into the system while the position of each parameter in the formula reveals the potential final effect on the magnitude of the respective increase or decrease. It must stress that the proposed methodology and the final mathematical solution are expressed in a quite general format. That means the whole discussion may be readily enhanced in many more control and input variables, and numerous sub-portfolios of loans. However, the dynamic hedging of each kind of risk separately, i.e. for particular credit risks, different repayment risks or regulatory risks etc. is not provided by this research. Moreover, another limit of the present model is the finite time horizon of control. Further research needs to overcome the above limits. The practical application provides compatible results with the respective theoretical analysis. These results show a steadily decreasing pattern for the investment mix towards more secure and liquid solutions – cash with the smaller standard deviation – in order to hedge the capital cost (Fig. 1). It must also be stressed the sensitively smaller supplementary lending rate as a result of a profitable investment policy and the existence of the accumulated surplus profit (Fig. 2 and Fig. 3).Finally, we should stress two more directions of further research. The first one considers the same problem with a generalization as regards the number of risky assets and consequently expansion of the number of the control variables. The second direction considers the construction and development of a structural, stochastic model in continuous time for the optimal risk management.