استراتژی مصون سازی ریسک تحت منحنی بازده یک تغییر مکان غیر موازیl

Under the assumption of the movement of rigid, a nonparallel-shift model in the term structure of interest rates is developed by introducing Fisher & Weil duration which is a well-known concept in the area of interest risk management. This paper has studied the hedge and replication for portfolio immunization to minimize the risk exposure. Throughout the experiment of numerical simulation, the risk exposures of the portfolio under the different risk hedging strategies are quantitatively evaluated by the method of value at risk (VaR) order statistics (OS) estimation. The results show that the risk hedging strategy proposed in this paper is very effective for the interest risk management of the default-free bond.

Duration and convexity are well-known concepts in gap management and asset-liability management in banks and other financial institutions. However, the simplicity that characterizes both concepts is based on an extreme simplification of interest rate movements, which are defined as parallel and instantaneous shifts in the term structure of interest rates. This unreliable assumption restricts the application of duration and convexity (see Ref. [1]). So, it is of great significance to design a risk hedging strategy under nonparallel shift of the yield curve. Recent years have seen many refinements in the duration models towards empirical multiple factor duration models which involve several duration measures that look for reality-based relations between interest rate changes. Such an empirical multiple factor duration models could be categorized into directional duration models, partial duration models and polynomial duration models. The first group, directional duration models, attempts to use movements in the term structure of interest rates to deduce the state variables, observable or not, which govern changes of the whole curve. Cooper [2] introduced a factor to capture the movement of the slope of yield curve, proposed a model that could reflect the variation of the term structure of interest rates. Litterman and Scheinkman [3] showed these three factors of level, steepness, and curvature as basic variables to control the variation of the yield curve, and presented a three-factor model that could describe the movement of term structure of interest rates. These models partially relaxed the parallel-shift assumption of the yield curve, and have broadened the application of duration and convexity. However, for evaluating interest rate risk exposure and hedging against this kind of risk, there are still some problems, the main ones being are the complexity of computation and poor accuracy of the result of computation (see Ref. [4]). Partial duration models explain interest rate changes by shifts in the level of the different segments into which the term structure is subdivided or by shifts in a limited number of interest rates. Ho [5] proposed the concept of key-rate durations to obtain a measure of the risk exposure of a single bond or a bond portfolio. The key-rate durations represent the price sensitivity to a change of each key rate. Further, Reitano [6] and [7], and Johnson and Meyer [8] assumed that interest rate changes by parallel shifts in the different segments, then transformed the problem of risk immunization under nonparallel shift of the yield curve into the one under parallel shift of the yield curve. The advantage of this segment methodology is obvious: the idea is simple, and the calculation process is realizable and practicable. But the most difficult problem in practice remains unresolved: how to choose the division of the term structure and the way to match different segments. Finally, polynomial duration models assume a polynomial fitting of term structure shifts. Nawalkha and Chambers [9] derive a polynomial bond return generating function without requiring that term structure shifts be expressed as a polynomial. Soto [10] built a polynomial duration model and did research on the Spanish government debt market. Other refined methods have been developed to handle nonparallel shifts of term structure of interest rates, such as value at risk (VaR). VaR is a modern way of financial risk management, which allows various immeasurable subjective factors to be transformed into probability of mathematical statistics. Such feature makes VaR have a far-reaching impact. Although VaR is an inadequate measure within the expected utility framework, it is at least as good as other traditional risk measures (see Ref. [11]). So the risk measurement of VaR is a challenging issue in modern statistics and applied mathematics. Until now, many scholars have investigated into this problem. There are lots of recent working papers in the well-known website about VaR, www.gloriamundi.org. Berkowitz [12] is a classical one in the application of VaR into risk management of commercial bank. Based on history data, this paper investigated the P&L distribution of the portfolios of six well-known American banks and characteristics of daily VaRs of those, which revealed the intrinsic reason of the instability of trading level in commercial banks. Based on the principle of rigid movement, this paper presents a general methodology in designing a risk hedging strategy under the nonparallel-shift yield curve, and has testified its validity and applicability by numerical simulation experiments and the technology of VaR order statistics (OS) estimation, the result of which can offer certain foundation for the practice of risk management in commercial banks and other financial institutions. This paper is organized as follows. In the second section, we construct an analysis model of rigid movement under unparallel shift of the yield curve. In Sections 3 and 4, we, respectively, introduce the general method to hedge the risk of interest rates under the yield curve both in parallel and nonparallel shift. Section 5 exhibits a general description of the methodology of VaR OS. Lastly, we calculate VaRs of different hedge portfolios and the related confidence intervals under certain scenario, and further analyses the validity of various risk hedging strategies.

In this paper, we adopt the bond's interest rate data of 1998, which came from America Federal Reserve, and select a note B1B1 (0.5 year), a bond B2B2 (10 year) and a bill B3B3 (20 year) as standard hedging instruments. Then we make a comparison of the results between the risk hedging strategy presented in this paper and that of simple duration. Suppose that the P/L of hedging portfolio is in t distribution. We can quantitatively analyze the risk of the bond portfolio by calculating each VaR and the corresponding confidence interval of various risk hedging strategies with the methodology of OS, so as to testify the validity of the risk hedging strategy under nonparallel shift of the yield curve. Investor's sample bonds are bond b1b1 (5 years), bill b2b2 (13 years) and bill b3b3 (15 years). We hedge the sample bonds using three standard hedging instruments B1B1, B2B2 and B3B3. The details of their cash are shown in Table 1.According to our risk hedging strategy, we know that the sample bill b3b3 (15 years) needs standard hedging instruments bill B3B3 and bond B2B2. From Eq. (9), we need standard hedging instruments bill B3B3$-59.505$-59.505 and bond B2B2$-47.181$-47.181 to hedge sample bill b3b3 $100. The details of standard hedging bond's position which investor should hold under different risk hedging strategies are shown in Table 2. The results of our risk hedging portfolios are illustrated in Fig. 2, Fig. 3 and Fig. 4 (Fig. 2 for sample bill b3b3 hedge, Fig. 3 for sample bill b2b2 hedge, and Fig. 4 for sample bond b1b1 hedge). For comparison, the P/Ls of the portfolios hedged only with one instrument based on the traditional duration hedge (Eq. (5)) are also shown. The portfolio hedged with only one instrument can be characterized as having dramatic P/L swings. The reason is that the change of long-term interest rate is different from that of short-term interest rate. Using the traditional duration hedge, although duration of the portfolio is matched, the changes of interest rates between the sample and instrument are different (small or great), so the volatility of the portfolio's value is very great. However, the portfolio hedged with our method includes two instruments: one term is shorter than the term of sample, the other term is longer than the term of sample. The influence of two instruments is counteracted. So the volatility of the portfolio's value is smaller than traditional hedging portfolio. The risk cannot be removed completely because of system risk. The results can be shown clearly in Figs. 2–4.For further analysis about the validity of various risk hedging strategies, given the confidence interval 97.5%, the paper employs the methodology of OS to calculate VaRs of various hedging portfolios, and also shows the corresponding confidence intervals. The results of the quantitative analysis of these hedging portfolios are shown in Table 2. The VaR of the sample bill b3b3 is -3.9080-3.9080. Using the traditional duration hedge, the VaR of the portfolio hedged with bill B3B3 is -1.3319-1.3319, the VaR of the portfolio hedged with bond B2B2 is -0.5502-0.5502. The interest rate risk of the portfolio is reduced. With our risk hedging strategy, the VaR of the portfolio is only -0.2579-0.2579 whose interest rate risk is the lowest. For the sample bond b2b2 and b1b1, the similar results are shown in Table 2. If there are many sample bonds, the interest rate risk of the portfolio hedged with our method is the lowest. One simple example is shown in Table 3All analysis methods indicate that, using our risk hedging strategy under nonparallel shift of the yield curve, we can gain a diversified portfolio, and then prevent and control the interest rate risk of default-free bonds effectively.