تجزیه و تحلیل حساسیت VAR و کمبود مورد انتظار برای اوراق بهادار تحت خالصسازی موافقت نامه ها

In this paper, we characterize explicitly the first derivative of the Value at Risk and the Expected Shortfall with respect to portfolio allocations when netting between positions exists. As a particular case, we examine a simple Gaussian example in order to illustrate the impact of netting agreements in credit risk management. Collateral issues are also dealt with. For practical purposes we further provide nonparametric estimators for sensitivities and derive their asymptotic distributions. An empirical application on a typical banking portfolio is finally provided.

For risk management purposes, the evaluation of marginal impacts of current or new positions on risk measures and regulatory capital has been recognized as an important point (Garman, 1996; Jorion, 1997). In practice, this evaluation can be made through explicit estimators of the first order derivatives of some standard risk measures, such as the Value at Risk (VaR) and the Expected Shortfall (ES), with respect to portfolio allocations (Gouriéroux et al., 2000, hereafter GLS; Scaillet, 2004). Knowledge of the sensitivity is helpful in reducing the amount of computational time needed to process large portfolios since it avoids the need to recompute risk measures each time the portfolio composition is slightly modified (Kurth and Tasche, 2003; Martin et al., 2001; Martin and Wilde, 2002). Besides it allows decomposing global portfolio risk component by component, and identifying the largest risk contributions (Denault, 2001; Garman, 1997; Hallerbach, 2003; Tasche, 1999). These derivatives are also of particular relevance in portfolio selection problem (see Markowitz (1952) for portfolio selection in a mean–variance framework). They help to characterize and evaluate efficient portfolio allocations1 when VaR and ES are substituted for variance as a measure of risk (GLS, 2000; Rockafellar and Uryasev, 2000; Yamai and Yoshiba, 2002b). In fact, numerical constrained optimization algorithms for computations of optimal allocations usually require consistent estimates of first order derivatives in order to converge properly. Unfortunately, the results available up to now have fallen short of tackling the problem of netting. Clearly, this is an important omission since most financial positions with respect to one or several counterparties are netted in practice. Neglecting the impact of netting will bias the evaluation of marginal impacts of current or new positions on risk measures and regulatory capital, and will lead to inefficient allocations in portfolio selection problems. Generally speaking, when trading partners agree to offset their positions or obligations, we say that there is netting. By doing so, they reduce a large number of positions or obligations to a smaller number of positions or obligations, and it is on this netted position that the two trading partners settle their outstanding obligations. In the financial community, positions are most of the time netted inside standardized juridical contracts. Streamlining of documentation has taken place as a result of joint efforts by regulators and financial industry organizations. In 1990, the Bank of International Settlements (BIS) issued minimum standards for the design and operation of netting schemes,2 while in 1991, the Federal Deposit Insurance Corporation Improvement Act (FDICIA) provided support for netting contracts among banks and other financial institutions. In 1992, the International Swaps and Derivatives Dealers (ISDA) issued its first version of the well-known “ISDA Master Agreement” for over-the-counter (OTC) derivatives markets. Its amended versions are still in force between most market participants around the globe today. To figure out the relative importance of OTC derivative markets, we recall that the total estimated notional amount of outstanding contracts stood at $141.7 trillion at the end-of-December 2002 (BIS, 2003), an 11% increase from end-of-June 2002. This compares with a 15% increase in the first half of 2002. At the same time, gross market values grew sharply, rising by 43% to $6.4 trillion. OTC business continued to accelerate significantly relative to that on exchanges.3 As mentioned previously, the term netting is used to describe the process of offsetting mutual positions or obligations e.g. to offset an obligation owed by bank A to bank B with an obligation owed by bank B to bank A. There are three main techniques for netting: • The payment or settlement netting is the process of settling all deals between two counterparties on a net cash basis, in the same currency. It can be informal or based on a formal agreement. The credit risk between counterparties remains unchanged and they stay legally obligated to settle the gross amounts of their positions. Payment netting allows for reducing the need for intra-day liquidity or credit in bridging the timing gaps between gross payments and gross receipts. • The netting by novation means that a single net amount is contractually substituted for previous individual gross sums owed between two counterparties, i.e. existing obligations are discharged by replacing them with a new obligation. Thus, netting by novation is a formal agreement that aims to reduce liquidity and counterparty credit risk. This novation process may take place automatically within the trading day, on the exchange of confirmations between the two banks. The claims need to be in the same currency. This process can be repeated an infinite number of times until the cut-off time for a particular settlement date. Although netting by novation is, in essence, a bilateral mechanism, it can be operated on multilateral basis within a larger group of banks. • The close-out netting is an arrangement to settle all contracted but not due liabilities to and claims on an institution by one single payment, immediately after the occurrence of a default or termination event.4 Since netting by close-out only operates upon the occurrence of a designated event, it cannot have any impact upon the number of payment messages passing between the counterparties in their normal trading relationship. Equally, it has no impact on liquidity or credit risk. If a termination event occurs, each trade is settled individually on due date unless the counterparties also agree on a supplementary netting by novation. Payment and close-out nettings are part of the ISDA Master agreement. The use of netting techniques can bring significant benefits for balance-sheet purposes, capital usage, credit risk and operational efficiencies.5 Indeed, it reduces the number of payment messages that have to be exchanged between counterparties. This lowers transaction costs and communication expenses, as well as the chance of mistakes. Moreover, netting is important because it reduces credit and liquidity risks, and ultimately systemic risk. Taking an offsetting position subject to a netting agreement is thus related to credit risk mitigation. The tendency of the regulator is to allow wider range of credit risk mitigants in order to avoid the so-called “domino effect” in the financial sector. In the new Basel Capital Accord (see the Basel Commission on Bank Supervision (2001) consultative document), on-balance sheet netting agreements of loans and deposits of banks to or from a counterparty will be permitted under some conditions. Note that, in 1995, the 1988 Basel Accord was modified to allow banks to reduce the credit exposures (“credit equivalent” in the Basel terminology) of their derivative positions by bilateral netting procedures (see Crouhy et al., 1998). The paper is organized as follows. In Sections 2 and 3 our aim is to extend the sensitivity analysis of VaR and ES to a setting in which netting is allowed, and to propose suitable estimators of the first order derivatives of VaR and ES in that context. We analyze both risk measures since pros and cons exist for each of them (see e.g. the papers of Yamai and Yoshiba, 2002a, Yamai and Yoshiba, 2002b, Yamai and Yoshiba, 2002c and Yamai and Yoshiba, 2002d). However, ES seems to benefit from clear advantages in a credit environment (Frey and McNeil, 2002). In Section 2, we outline the framework and explain the differences arising from netting agreements in case of default of a given counterparty. The loss function associated with netted positions is no more a simple sum of exposures or mark-to-market valuations, but rather involves some nonlinearities. More precisely, it involves some terms like (Y1 + ⋯ + YI)+ = max(Y1 + ⋯ + YI, 0), when the positions Yi, i = 1, … , I, belong to the same netting agreement at default. Section 2 contains the main result of the paper, namely the explicit characterization of the first order derivatives of VaR and ES for portfolios under netting agreements. The Gaussian case is briefly developed for illustrative purposes. In particular we compare VaR and its sensitivity in the netted and unnetted cases, and show on a simple stylized example that netting is a valuable credit risk mitigation device. In Section 3 we outline an extension of the approach, which allows for the presence of collateral. In Section 4, we derive estimators of sensitivities so that they can be used in practical risk management and portfolio selection procedures. These estimators are of a nonparametric nature and easy to implement. In Section 5 we provide an empirical illustration for a typical portfolio of a large bank. Section 6 contains some concluding remarks. Technical appendices gather proofs.

Risk measures answer the need of quantifying the risk of potential losses on a portfolio of assets. This need may arise due to internal concerns (risk-reward tradeoff) or external constraints (prudential rules imposed by regulators). In this paper we have proposed estimation procedures allowing for a sensivity analysis w.r.t. changes in the portfolio allocation. The setting explicitly takes into account the possibility of netting and collateral agreements. The estimation procedures are nonparametric, fast and easy to implement. They have also been shown to be of practical relevance in real life banking situations, and should help to achieve better credit risk management in the future.