مدیریت ریسک بدهی های بدون سر رسید

Risk management of non-maturing liabilities is a relatively unstudied issue of significant practical importance. Non-maturing liabilities include most of the traditional deposit accounts like demand deposits, savings accounts and short time deposits and form the basis of the funding of depository institutions. Therefore, the asset and liability management of depository institutions depends crucially on an accurate understanding of the liquidity risk and interest rate risk profile of these deposits. In this paper we propose a stochastic three-factor model as general quantitative framework for liquidity risk and interest rate risk management for non-maturing liabilities. It consists of three building blocks: market rates, deposit rates and deposit volumes. We give a detailed model specification and present algorithms for simulation and calibration. Our approach to liquidity risk management is based on the term structure of liquidity, a concept which forecasts for a specified period and probability what amount of cash is available for investment. For interest rate risk management we compute the value, the risk profile and the replicating bond portfolio of non-maturing liabilities using arbitrage-free pricing under a variance-minimizing measure. The proposed methodology is demonstrated by means of a case study: the risk management of savings accounts.

1.1. Importance of non-maturing liabilities in the AL management The asset/liability problem that depository institutions face is quite simple to explain – although not necessarily easy to solve. A depository institution seeks to earn positive spread between the assets it invests in (loans and securities) and the cost of its funds (deposits and other sources). The spread income should allow the institution to meet operating expenses and earn fair profit on its capital. In generating spread income a depository institution faces several risks, most important credit risk, interest rate (or funding) risk and liquidity risk. The management of interest rate and liquidity risk is particularly difficult for non-maturing liabilities, i.e. deposits without a specific maturity or deposits whose actual time horizon significantly differs from their contractual maturity. Non-maturing liabilities include most of the traditional deposit accounts like demand deposits, savings accounts and short time deposits and form the basis of the funding of depository institutions. Therefore, the asset and liability management of depository institutions depends crucially on an accurate understanding of the liquidity risk and interest rate risk profile of these deposits. For quite some time bank regulators have been aware of the significance of proper risk management for non-maturing liabilities but also of the difficulties. In fact, uncertainty about measuring interest rate risk for these deposits was one factor in not adopting formal interest rate capital guidelines for banks' non-trading positions (see Federal Reserve, 1995; O'Brien, 2000). Recently several consulting papers proposed the introduction of a capital charge for institutions with a high exposure to interest rate risk arising from banking book items like non-maturing deposits (BIS, 1999; EU, 1999). These new regulations will increase the pressure on depository institutions to use transparent quantitative methodologies for the risk management of these positions. 1.2. Quantitative risk management techniques The central problem in the risk management of non-maturing liabilities is the assignment of a maturity profile to these liabilities or, equivalently, the construction of a replicating bond portfolio with fixed maturities. Many banks use the following approach: 1. First the core of a balance position is determined, then the floating part is defined as balance minus core. 2. The floating part is invested in the overnight (O/N) bucket. The core is sub-divided into portions which are invested in different time bands. Maturing tranches are reinvested in the same time band. The subdivision into floating part and core portions with different maturities is usually done in a rather arbitrary way without theoretical justification. In fact, despite its significance this problem is still relatively unstudied. In the last ten years a number of papers on managing non-maturing assets and liabilities have been published including Ausubel (1991), O'Brien et al. (1994), the Office of Thrift Supervision (1994), Hutchinson and Pennacchi (1996), Selvaggio (1996), Jarrow and van Deventer (1998), Schürle (1998), Janosi et al. (1999) and O'Brien (2000). Hutchinson and Pennacchi (1996) value non-maturing liabilities in an equilibrium-based model. Jarrow and van Deventer (1998) propose an arbitrage-free approach for valuation and hedging. They show that non-maturing liabilities are equivalent to particular interest rate swaps. They obtain an analytic valuation formula in a simple one-factor model with deposit rates and volumes given by deterministic functions of the short rate and the short rate specified by an extended one-factor Vasicek model. Janosi et al. (1999) provide an empirical investigation of the Jarrow and van Deventer model in the US market. O'Brien (2000) focuses on the US market as well. In his valuation model deposit rates and balances are represented by autoregressive processes. Alternative deposit rate specifications studied include asymmetric adjustment to market rate changes. Schürle (1998) applies stochastic optimization techniques to the margin optimization problem for non-maturing liabilities (see also Frauendorfer and Schürle, 2000). The objective of our paper is the development of a general quantitative framework for liquidity risk and interest rate risk management for non-maturing liabilities. Our model consists of three building blocks. Market rates: The market rate model serves as framework for valuing the cash flows of the non-maturing liabilities specified by deposit rates and volumes. In contrast to derivative pricing, our main focus is a realistic development of interest rates over a long period of time and not the exact fit to current market prices of plain vanilla instruments. We therefore use historical time-series for calibration. We have currently implemented two classes of two-factor HJM models: two-factor Vasicek models and non-parametric HJM models with piecewise constant volatility functions.1 The calibration of both models is based on principal component analysis. In particular for the non-parametric model we have obtained very good results (Kalkbrener, in preparation). Deposit rates: Deposit rates are heavily influenced by market rates but rates of different types of deposits differ significantly. In particular, sensitivities to interest rate changes vary. We therefore propose the following general concept for modeling deposit rates: deposit rates are given by a deterministic function with only the market rates as stochastic arguments, no additional stochastic factor is used. Market rates of different maturities can be used as arguments and – even more important – no restrictions are made concerning the form of the deterministic function. Deposit volumes: Our analysis shows that correlations between deposit volumes and market rates are not particularly high in the German market. We therefore introduce an additional stochastic factor for deposit volumes. This factor may be correlated to the two factors of the market rate model. We have currently implemented two diffusion models for deposit volumes, a normally and a lognormally distributed model. By combining the three components we obtain a three-factor model for non-maturing liabilities which we use as framework for risk management. Liquidity risk management: Because of their high volumes and their characteristics non-maturing liabilities are of great importance for the liquidity risk management in depository institutions. The central problem is the following: what amount of cash is available for investment over a given time horizon [0,t] with a given probability p? In order to answer this question we introduce the following simple concept. Let V(u) be a stochastic process which specifies the volume of deposit accounts and define the process of minima by

In this paper we have presented a general quantitative framework for liquidity risk and interest rate risk management for non-maturing liabilities. Our approach to liquidity risk management is rather straightforward: we derive a stochastic process for deposit volumes from a historical time series and compute the term structure of liquidity. This term structure gives the probabilities that the volume drops below specified levels in specified time intervals and is used to immunize a replicating portfolio against liquidity risk. Deposit volumes are either defined as Ornstein–Uhlenbeck processes with linear trend or exponential functions of those processes. Despite its simplicity the model produced reliable results for the deposit volumes we worked with. Nevertheless it seems worthwhile to experiment with more realistic volume models, for instance models based on extreme value theory (Embrechts et al., 1997). Another approach to modeling deposit volumes is to investigate correlations to macro economic variables. The analysis of the dependence of customer behaviour on the macro economic environment can certainly give important insights. However, it seems questionable whether more reliable volume forecasts can be obtained in this way since forecasting macro economic developments over longer periods is not a trivial task. The interest rate risk management techniques we propose are based on arbitrage-free valuation. This methodology is transparent and consistent with the current practice of derivatives pricing and with the risk management methods for the trading book. Furthermore, our portfolio replication methodology is in line with suggestions made by regulatory authorities. In this paper we have focused on the construction of replicating bond portfolios. However, the methodology can be easily extended to portfolios which contain not only bonds but also derivatives like caps and floors. Further analysis and comparison to the techniques suggested by Thomas Ho (Ho, 1992; Ho and Chen, 1995) are important topics for future research. An interesting alternative to arbitrage-free pricing are risk management techniques based on stochastic optimization. These techniques seem to be a natural approach to a number of financial planning problems, for instance to investment problems in asset and liability management. Stochastic optimization models not only reflect the uncertainty in the future development of risk factors. They also provide a framework for modeling different investment strategies over the entire planning period and techniques for determining the optimal solution. The main obstacle in applying stochastic optimization to realistic investment problems is the complexity of this approach. In general, large optimization problems have to be solved if risk factors and cash flows are modeled in a realistic way. Recently, stochastic optimization techniques have been applied to margin optimization for non-maturing liabilities (Schürle, 1998; Frauendorfer and Schürle, 2000). It seems promising to adapt these techniques to the portfolio replication problem considered in this paper. Our model for interest rate risk management of non-maturing liabilities consists of three building blocks: market rates, deposit rates and deposit volumes. Market rates are specified by a two-factor Vasicek or non-parametric HJM model. In our tests the non-parametric model outperformed the Vasicek model and met the requirement of realistic interest rates over long time periods rather well (Kalkbrener, in preparation). The disadvantage of the non-parametric model is its complexity: since the volatility functions do not have analytic form simulation of interest rate paths is time-consuming. A comparison to affine models (Brown and Schaefer, 1994; Duffie and Kan, 1996; Dai and Singleton, 1998) is planned. The value of deposit accounts and the structure of the replicating portfolio heavily depend on the specific form of the deposit rate process. In our model, no restrictions are imposed on the form of the deterministic function which specifies a deposit rate. We also allow market rates of different maturities as stochastic arguments. Our objective is to provide a framework which is flexible enough for modeling different types of deposit rates. The actual process definitions for specific types of deposit rates should to be based on a careful market analysis: problems like the dependence on market rates of different maturities or the asymmetric response to market rate changes, i.e. whether deposit rates move more rapidly when market rates drop than when they rise, deserve a closer study. This type of analysis is beyond the scope of the paper.