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Abstract:
The term structure of interest rates gives the relationship between the yield on an investment and the term to maturity of the investment. Since the term structure is typically measured using default-free, continuously-compounded, annualized zero-coupon yields, it is not directly observable from the published coupon bond prices and yields. This paper focuses on how to estimate the default-free term structure of interest rates from bond data using three methods: the bootstrapping method, the McCulloch cubic-spline method, and the Nelson and Siegel method. Nelson and Siegel method is shown to be more robust than the other two methods. The results of this paper can be implemented using user-friendly Excel spreadsheets.

interest rates, term structure, bonds, fixed income, Excel

Abstract:
This paper evaluates the performance of a kind of interest rate model that has increasingly been attracting the attention of the financial industry in recent years and which relies on principal component analysis to extract risk factors. Focusing on the Spanish bond market, our empirical analysis reveals that interest rate movements can be summarized by three principal components, related to the level, the steepness and the curvature of the yield curve. This three-principal component model is able to offer a balanced explanation of interest rate shocks and bond returns across maturities and overcomes typical one- and two-factor interest rate models. However, our results also reveal some variations with time in the principal components that point to the need to recognize the dynamic volatility structure of interest rates.

interest rate, bond, risk management, factor model, principal component

Abstract:
This paper shows how to price American interest rate options under the exponential jumps-extended Vasicek model, or the Vasicek-EJ model. We modify the Gaussian jump-diffusion tree of Amin [1993] and apply to the exponential jumps-based short rate process under the Vasicek-EJ model. The tree is truncated at both ends to allow fast computation of option prices. We also consider the time-inhomogeneous version of this model, denoted as the Vasicek-EJ model that allows exact calibration to the initially observable bond prices. We provide an analytical solution to the deterministic shift term used for calibrating the short rate process to the initially observable bond prices, and show how to generate the jump-diffusion tree for the Vasicek-EJ model. Our simulations show fast convergence of European option prices obtained using the jump-diffusion tree, to those obtained using the Fourier inversion method for options on zero-coupon bonds (or caplets), and the cumulant expansion method for options on coupon bonds (or swaptions).

Bond options, Interest Rate Trees, Jumps, Vasicek Model, American options

Abstract:
Dynamic Term Structure Modeling, the second book in the trilogy on the Fixed Income Valuation Course, shows how to value interest rate derivatives and credit derivatives using a variety of affine, quadratic, HJM, and LIBOR market models. Using a new taxonomy, this book classifies all term structure models as either fundamental models, or preference-free single-plus, double-plus, and triple-plus models. Filled with in-depth insights and expert advice, this book shows how to price basic interest rate and credit derivative products, such as Treasury and Eurodollar futures, bond options, interest rate options (e.g., caps, floors, and swaptions), forward rate agreements, interest rate swaps, credit default swaps, credit spread options, and others. With intuitive explanations and fully developed examples, this book provides new transforms for building efficient trees under state-dependent volatility models, stochastic volatility models, and jump-diffusion models, for pricing American options, and describes fast computational methods such as the Fourier inversion method (including the FFT) and the cumulant expansion method, for valuing interest rate derivatives and credit derivatives, under a variety of affine, quadratic, and LIBOR market models. This book is also accompanied by an informative CD-ROM, which contains various Excel®/VBA® spreadsheets. This software allows for valuation of interest rate derivatives by building interest rate trees for low-dimensional affine models, as well as computing solutions using quasi-analytical formulas for higher-dimensional affine, quadratic, and LIBOR market models. Though most of the programs require coding in advanced scientific languages, such as C and C++, the final output is presented in user-friendly Excel/VBA spreadsheets.

term structure models, caps, swaptions, credit default swaps, credit derivatives

Abstract:
How do the managers of financial institutions hedge against the effects of non-parallel yield curve shifts? This paper addresses this important issue by reviewing the important findings in the area of interest rate risk management over the past two decades. We discuss four classes of models in the fixed income literature that deal with hedging the risk of large, non-parallel yield curve shifts. These models are given as M-Absolute/M-Square models, duration vector models, key rate duration models, and principal component duration models. These models can be used for designing various passive strategies such as portfolio immunization, bond index replication, and duration gap management; and hybrid strategies (i.e., active/passive) such as targeted yield curve shifts speculation (based on change in either the height, and/or the slope, and/or the curvature of the yield curve) and contingent immunization.

interest rate, yield curve, fixed income, duration, immunization, portfolio strategy

Abstract:
Are the managers of financial institutions ready for the small but increasingly significant risk of inflation in the near future, due to the unprecedented fiscal and monetary responses of the U.S. government to prevent an economic collapse? This paper addresses this important issue by reviewing important findings in the area of interest rate risk management. We discuss five classes of models in the fixed income literature that deal with hedging the risk of large, non-parallel yield curve shifts. These models are given as M-Absolute/M-Square models, duration vector models, key rate duration models, principal component duration models, and extensions of these models for fixed income derivatives, for valuing and hedging bonds, loans, demand deposits, and other fixed income instruments. These models can be used for designing various hedging strategies such as portfolio immunization, bond index replication, duration gap management, and contingent immunization, to protect against changes in the height, slope, and curvature of the yield curve. We argue that the current regulatory models proposed by the U.S. Federal Reserve, the Office of Thrift Supervision, and the Bank of International Settlements, may understate the true interest rate risk exposure of financial institutions, if sharp increases in interest rates lead to higher default risk and quickening of the pace of deposit withdrawals.

interest rate risk, duration, convexity, key rate, inflation

Abstract:
This paper presents jump extensions to the Cox, Ingersoll, and Ross (CIR) and the constant-elasticity-of-variance (CEV) models of the short rate, with analytical solutions for the case of exponential jumps, and efficient lattice-based solutions for both exponential jumps and lognormal jumps. We demonstrate how to superimpose a recombining multinomial jump tree on the diffusion tree, creating the mixed jump-diffusion trees for CIR and CEV models extended with jumps. Finally we also present the preference-free versions of these models that allow these models to be fully calibrated to an initially observed forward rate curve, making them consistent with the HJM [1992] paradigm. Our simulations show fast convergence of the trees to the respective analytical solutions.

Interest rate models, Term structure models, Jumps, CIR, CEV, Trees

Abstract:
This paper generalizes the M-square and M-vector models (Fong and Fabozzi [1985] and Nawalkha and Chambers [1997]) by using a Taylor series expansion of the bond return function with respect to simple polynomial functions of the cash flow maturities. The classic M-vector computes the weighted averages of the distance between the maturity of each cash flow and the portfolio horizon, raised to integer powers (e.g., (t - H)^1, (t - H)^2, (t - H)^3, etc.). Implementation of the new approach involves computing the weighted averages of the distance between some polynomial function of the maturity of each cash flow and that of the portfolio horizon, raised to integer powers (e.g., (t^0.5 - H^0.5)^1, (t^0.5 - H^0.5)^2, (t^0.5 - H^0.5)^3, etc.). We test six different generalized M-vector models corresponding to six different polynomial functions. It is shown that polynomial functions of lower power (i.e., 0.25 or 0.5) provide significantly enhanced protection from interest rate risk, when higher-order generalized M-vector models are used.

immunization, duration, interest rate, risk management, fixed income

Abstract:
This paper gives a new taxonomy of dynamic term structure models that classifies all existing TSMs as either fundamental models or preference-free single-plus, double-plus, and triple-plus models. We exemplify the new taxonomy by considering preference-free versions of some well-known fundamental short rate models. Single-plus extensions of the fundamental models are shown to be both time-homogeneous and preference-free - two characteristics which do not simultaneously hold under any existing class of TSMs. Though the analytical apparatus for pricing fixed income securities is identical under fundamental models and single-plus models, the latter models are consistent with general non-linear forms of MPRs which may also depend upon an arbitrary set of state variables, leading to better estimates of risk-neutral parameters. The preference-free double-plus and triple-plus extensions of the fundamental models are similar to the Heath, Jarrow, and Morton [1992] models, in that time-inhomogeneous drifts and volatilities are used as "smoothing variables" to fit the initial bond prices and initial term structure of volatilities, respectively.

Term Structure, Interest Rate, Single-Plus, Double-Plus, Vasicek, CIR, USV

Abstract:
This paper carries out a comprehensive analysis of the interest rate risk borne by the Spanish firms on a sector basis. The traditional linear interest rate exposure model has been extended to allow for the possibility of a nonlinear exposure component as well as the presence of asymmetric behaviour in the exposure pattern. The obtained results show a significant interest rate exposure for some sectors, especially with regard to changes in the long-term interest rates. Moreover, it is documented that the linear exposure profile prevails over the asymmetric and nonlinear exposure patterns. In particular, the Construction sector is the sector that shows the highest incidence of interest rate risk in the Spanish case.

interest rate exposure, stock, firm, sector, risk management

Abstract:
In this paper, we analyze the influence of portfolio design on the goal of immunization in Spanish bond portfolios. Extending the work of Fooladi and Roberts (1992) and Bierwag et al. (1993), we test a wide set of strategies which includes duration-matching strategies and strategies based on the M-squared of Fong and Vasicek (1984) and the M-absolute of Nawalkha and Chambers (1996). We attempt to evaluate the effectiveness of these dispersion measures and justify the improvements in immunization when portfolios include a bond maturing near the horizon date.

immunization, duration, interest rate, risk management, fixed income

Abstract:
This paper focuses on the Spanish government debt market in an attempt to evaluate the immunization performance of the polynomial duration model of Chambers and Carleton (Chambers, D.R., Carleton, W.T., 1988. A generalized approach to duration. In: Chen, A.H. (Ed.), Research in Finance, vol. 7, JAI Press, Greenwich, pp. 163-181), in default-free and option-free fixed-income portfolios and to ascertain whether traditional convexity is an earnings-generating element. Empirical tests show that three constraints, namely those related to the level, slope and curvature of term structure shifts, are necessary to guarantee a return close to the target. The only exception to this rule is found in portfolios including an asset that matures near the horizon date, in which classical immunization performs properly.

Immunization, Duration, Convexity, Portfolio management, Term structure

Abstract:
This paper compares the immunization performance of alternative single and multiple factor duration models, using Spanish government bond data, over 1, 2 and 3-year horizons. The aim is to assess whether the success of duration-matching strategies is primarily attributable to the particular model chosen or to the number of risk factors considered. Empirical tests show that: (i) traditional immunization is easily bettered by more realistic strategies; (ii) the number of risk factors considered has a greater influence on the result than the particular model chosen; and (iii) three-factor immunization strategies offer the highest immunization benchmarks.

Immunization, Duration, Interest rate, Risk management, Fixed income

Abstract:
This paper derives analytical solutions for valuing credit default swaps (CDS) using preference-free multifactor affine and quadratic models, under the recovery of face value (RFV) assumption. We use a preference-free framework, which is independent of the market prices of risk, and yet allows the short rate process and the spread process to be time-homogeneous. The solutions allow arbitrary number of factors for the short rate and the default intensity, and nest the solutions of Longstaff, Mithal, and Neis [2003], and Pan and Singleton [2005]. The multifactor framework used in this paper allows a better fit with default-free bond prices and CDS spreads.

credit default swaps, CDS, reduced form models, interest rate models, term structure models, affine, quadratic

Abstract:
This paper derives analytical solutions for valuing Eurodollar/Euribor futures using multifactor affine and quadratic models. We use a preference-free framework independent of the market prices of risk. The preference-free "single-plus" models allow the short rate process to be time-homogeneous. The preference-free "double-plus" models allow the short rate process to be time-inhomogeneous, such that the model prices are consistent with initially observable bond prices. Our solutions allow arbitrary number of factors for the short rate and nest virtually all other solutions given in the literature. We also solve the convexity-bias in closed-form under various multifactor affine and quadratic models.

Eurodollar futures, Euribor futures, Interest rate models, Term structure models, Affine, Quadratic, Convexity bias