Читать книгу The Advanced Fixed Income and Derivatives Management Guide - Saied Simozar - Страница 10

One of the keys to managing investment portfolios is identification and measurement of sources of risk and return. In fixed income, the most important source is the movement of interest rates. Even though changes in interest rates at different maturities are not perfectly correlated, diversifying a portfolio across the maturity spectrum will not lead to interest rate risk reduction. In general, a portfolio of one security that matches the duration of a benchmark tends to have a lower tracking error with the benchmark than a well-diversified portfolio that ignores duration.

Historically, portfolio managers have used Macaulay or modified duration to measure the sensitivity of a portfolio to changes in interest rates. With the increased efficiency of the markets and clients' demands for better risk measurement and management, several approaches for modeling the movements of the term structure of interest rates (TSIR) have been introduced.

A few TSIR models are based on theoretical considerations and have focused on the time evolution or stochastic nature of interest rates. These models have traditionally been used for building interest rate trees and for pricing contingent claims. For a review of these models, see Boero and Torricelli [1].

Another class of TSIR models is based on parametric variables, which may or may not have a theoretical basis, and their primary emphasis is to explain the shape of the TSIR. An analytical solution of the theoretical models would also lead to a parametric solution of the TSIR; see Ferguson and Raymar for a review [2]. Parametric models can be easily used for risk management and they almost always lead to an improvement over the traditional duration measurement. Willner [3] has applied the term structure model proposed by Nelson and Siegel [4] to measure level, slope and curvature durations of securities.

Key rate duration (KRD) proposed by Ho [5] is another attempt to account for non-parallel movements of the TSIR. A major shortcoming of KRD is that the optimum number and maturity of key rates are not known, and often on-the-run treasuries are used for this purpose. Additionally, key rates tend to have very high correlations with one another, especially at long maturities, and it is difficult to attach much significance to individual KRDs. The most important feature of KRD is that the duration contribution of a key rate represents the correct hedge for that part of the curve.

Another approach that has recently received some attention for risk management is the principal components analysis (PCA) developed by Litterman and Scheinkman [6]. In PCA, the most significant components of the yield curve movements are calculated through the statistical analysis of historical yields at various maturities. A very attractive feature of principal components, as far as risk management is concerned, is that they are orthogonal to each other (on the basis of historical data). The first three components of PCA usually account for more than 98  % of the movements of the yield curve.

Another class of yield curve models is based on splines. Cubic splines are widely used for fitting the yield curve and are useful for valuation purposes, to the extent that the yield curve is smooth. Cubic splines can be unstable, especially if the number of bonds is relatively low. For a review of different yield curve models, see Advanced Fixed Income Analysis by Moorad Choudhry [7].

All of the above models are useful either for risk management or pricing, but not for both. For portfolio management applications, it is quite difficult to translate either KRDs or PCA durations into positions in a portfolio. Likewise, it is not straightforward to convert valuations from a cubic spline curve into risk metrics. For global portfolios, it would be impossible to compare the relative value of securities or the cheapness/richness of the areas of global yield curves using KRDs, PCA or cubic splines. Each currency requires a separate PCA, which in turn requires the availability of historical data.

In this book we will develop a market driven framework for fixed income management that addresses all aspects of fixed income portfolio management, including risk measurement, performance attribution, security selection, trading, hedging and analysis of spread products. For risk management, the model is as accurate as PCA and its first three components are very similar to those of PCA. For trading and hedging, the model can be easily transformed into KRDs. This framework has been successfully applied to the management of global portfolios, risk measurement and management, credit and emerging markets securities, derivatives, mortgage bonds and prepayment models, and for the construction of replicating portfolios.

The movements of interest rates are decomposed into components that are weakly correlated with each other and can be viewed as independent and diversifying components of a fixed income portfolio strategy. These interest rate components can be viewed as different sectors of the treasury curve. However, TSIR components tend to be more weakly correlated with one another in the medium term horizon than typical sectors of the equity market and therefore can offer better diversification potential.

First, we develop a parametric term structure model that can price the treasury curve very accurately. The model is highly flexible and stable and its movements are very intuitive. The components of the model represent the modes of fluctuations of the yield curve, namely, level, slope, bend etc. and in well behaved markets all bonds can be priced with an average error of less than 2 bps. The components of the yield curve or the basis functions, as we call them, can be converted to other basis functions such as Key Rate components. We will also compare the components of our model to PCA and to an economic indicator.

The model is then applied to risk measurement and management for treasuries. The components of the term structure directly translate into trades that fixed income practitioners are accustomed to such as bullets, barbells and butterfly trades of the yield curve. The level duration of a portfolio measures the net duration or bullet duration, while the slope duration measures the barbell strategies and bend duration measures the butterfly strategies. We also compare historical data using different basis functions.

In the performance attribution section, we show that the performance of a treasury portfolio can be measured with an accuracy of less than 1 basis point per year, by decomposing performance yield, duration and convexity and security selection components. We will further delineate the difference between various representations of the yield curve and provide some evidence associated with the weaknesses of Key Rate basis functions.

A few characteristics of the TSIR model are as follows:

● It is driven by current market prices and accurately prices treasuries using only five parameters.

● Risk measurement and portfolio replication do not require a historical correlation matrix for a country where the information is not available.

● Risk management, valuation, performance attribution and portfolio management can be integrated.

● It can be easily expanded if a higher number of components are desired without changing the value of primary components significantly.

● It is intuitive, is easy to use, implement and manipulate. Its components are readily identified with portfolio positions of duration, flattening/steepening, butterfly, etc.

● It is flexible and can be easily applied to mortgage prepayment models, emerging markets, multi-currency portfolios, inflation linked bonds, derivatives analysis, etc.

● It can be used as an indicator of relative value or relative curve positions in a consistent way across currencies and credits.

● The model is easily applied to all global rates, term structure of Libor, term structure of real rates and term structure of credit rates.

● The model is very stable and, unlike cubic splines, can be easily differentiated multiple times if necessary.

Throughout this book we have provided detailed examples of the applications of our model to risk measurement, performance attribution and portfolio management. We first introduce the concept of linear and non-linear time space and then construct the components of our term structure model and forward rates. Next, we derive duration and convexity components and calculate performance attribution from duration components.

In Chapter 6 Libor and interest rate swaps are covered and the model is applied to the term structure of Libor rates. It is shown that interest rate swaps have a structural problem that makes them subject to arbitrage. In Chapters 7 and 8 trading and portfolio optimization and security selection are examined. In Chapter 9 a model for the term structure of volatility surface is developed, and in Chapter 10 the effects of convexity and volatility on the shape of the TSIR are analyzed and the convexity adjusted TSIR model is developed. The convexity adjustment to eurodollar futures is also covered and potential arbitrage opportunities are pointed out. In Chapter 11 there is a very detailed and precise coverage of inflation linked bonds along with the application of the term structure of real rates to global inflation linked bonds as well as inflation swaps.

In Chapter 12 credit securities are analyzed and the term structure of credit rates (TSCR) with its application to performance attribution and risk measurement is analyzed. In Chapter 13 default and recovery or cash flow guarantees of credit securities are analyzed and for the first time the TSCR is used to estimate the market implied recovery rate. The application of the TSCR to credit default swaps and construction of performance attribution for complex portfolios are also analyzed in this chapter.

Analysis of global bond futures and their hedging, replication, arbitrage and performance attribution are covered in Chapter 14. Bond options and callable bonds are covered in Chapter 15 along with a very detailed analysis of American bond options with accuracy approaching closed form solutions. The weaknesses of the Black-76 model are pointed out and the model is applied to corporate bond options and exotic securities. It is shown that credit bond prices cannot follow the efficient market hypothesis and there are long term opportunities in the credit markets for fund managers.

In Chapter 16 currencies as an asset class along with their options and futures are covered and models to take advantage of currencies in a portfolio are explored. Chapters 17 and 18 cover the application of the TSIR to prepayments and development of mortgage analysis. In Chapter 19 product design and portfolio construction are covered and a method is developed to analyze the competitive universe of a bond fund. Chapter 20 covers detailed mathematical derivations of the parameters of the TSIR and TSCR and estimation of recovery value, and Chapter 21 covers implementation notes and short-cuts.