ABSTRACT Title of Dissertation: THREE ESSAYS ON MORTGAGE BACKED SECURITIES: HEDGING INTEREST RATE AND CREDIT RISKS Jian Chen, Doctor of Philosophy, 2003 Dissertation directed by: Professor Michael C. Fu The Robert H. Smith School of Business This dissertation includes three essays on hedging the interest rate and credit risks of Mortgage-Backed Securities (MBS). Essay one addresses the problem of how to efficiently estimate interest rate sensitivity parameters of MBS. To do this in Monte Carlo simulation, we derive perturbation analysis (PA) gradient estimators in a general setting. Then we apply the Hull-White interest rate model and a common prepayment model to derive the corresponding specific PA estimators, assuming the shock of interest rate term structure takes the form of a trigonometric polynomial series. Numerical experiments comparing finite difference (FD) estimators with our PA estimators indicate that the PA estimators can provide better accuracy than FD estimators, while using much lower computational cost. Using the estimators, we analyze the impact of term structure shifts on various mortgage products. Based these analysis, we propose a new product to mitigate interest rate risk. Essay two addresses the problem of how to measure interest rate yield curve shift more realistically, and how to use these risk measures to hedge the interest rate risk of MBS. We use a Principal Components Analysis (PCA) approach to analyze historical interest rate data, and acquire the volatility factors we need in Heath-Jarrow-Morton interest rate model simulation. Then we propose a hedging algorithm to hedge MBS, based on PA gradient estimators derived upon these PCA factors. Our results show that the new hedging method can achieve much better hedging efficiency than traditional duration and convexity hedging. Essay three addresses the application a new regression method on credit spread data. Previous research has shown that variables in traditional structural model have limited explanatory power in credit spread regression. We argue that this is partially due to the non-constancy of the credit spread gradients to state variables. We use a Random Coefficient Regression (RCR) model to accommodate this problem. The explanatory power increases dramatically with the new RCR model, without adding new independent variables. This is the first work to address the dependence between credit spread sensitivities and state variables of structural in a systematic way. Also our estimates are consistent with prediction from Merton?s structural model. THREE ESSAYS ON MORTGAGE BACKED SECURITIES: HEDGING INTEREST RATE AND CREDIT RISKS By Jian Chen Dissertation submitted to the Faculty of the Graduate School of the University of Maryland in partial fulfillment of the requirements for the degree of Doctor of Philosophy November 2003 Advisory Committee: Professor Michael Fu, Chair Professor Dilip Madan Professor Haluk Unal Professor Nengjiu Ju Professor Eric Slud ?Copyright by Jian Chen 2003 ii Dedicated to Huixian Jackie Xu, my lovely wife Li Xueqing and Chen Zhixuan, my parents Chuan Chen, my son iii ACKNOWLEDGMENTS I would like to sincerely thank my advisor, Dr. Fu, who admitted me six years ago, and has arduously helped me since then, especially during the lengthy process of my dissertation writing. I also would like to thank Dr. Unal, for providing me most valuable advice for my last essay. I also would like to thank Dr. Madan, Dr. Ju, and Dr. Slud for being my committee members and providing a lot of insightful comments on the dissertation proposal. A significant portion of this dissertation is completed during my tenure at Fannie Mae. I benefited a lot from conversations with my colleagues there. I would like to thank Dr. Yigao Liang for his inputs on Essay I. I would like to thank Dr. Alex Philipov and Dr. Arash Sotoodehnia for discussion on HJM model. I want to thank Ms. CJ Zhao on credit spread discussion. I also want to thank Dr. Jay Guo, who helped me a lot for the last essay. Also I would like to thank Dr. Levant Guntay of Indiana University, who kindly provided some data for the last essay, and has given me many research ideas during our discussions. Needless to say, all errors are mine own. All opinions expressed in this dissertation are not Fannie Mae?s but mine own. iv TABLE OF CONTENTS List of Figures vi List of Tables x List of Abbreviations xi 1. Introduction 1 1.1 Efficient Sensitivity Analysis of Mortgage Backed Securities 3 1.2 Hedging MBS in HJM Framework 9 1.3 Hedging Credit Risk of MBS: A Random Coefficient Approach 10 2. Efficient Sensitivity Analysis of Mortgage Backed Securities 12 2.1 Problem Setting 9 2.2 Derivation of General PA Estimators 14 2.2.1 Gradient Estimator for Cash Flow 15 2.2.2 Gradient Estimator for Discounting Factor 19 2.3 Applying the Gradients 20 2.3.1 Interest Model Setup 20 2.3.2 Trigonometric Polynomial Shocks 23 2.3.3 Derivation of Gradients with respect to Modified Fourier Series 27 2.3.4 Derivation of Gradients with respect to Volatility: Vega 31 2.3.5 Derivation of Second Order Gradients: Gamma 31 2.3.6 Derivation of ARM estimators 35 2.4 Numerical Example 41 2.4.1 Specification of Numerical Example 41 v 2.4.2 Comparison of PA and FD estimators 42 2.4.3 Result Analysis 53 2.5 Interpretation of the Results 60 2.5.1 Overview of the Results 60 2.5.2 Modified Fourier Shock Impact 62 2.5.3 Potential New Product 70 2.6 Conclusion 75 3. Hedging MBS in HJM Framework 77 3.1 Motivation 77 3.2 Estimation of Volatility Factors via PCA 80 3.3 Simulation in HJM Framework 84 3.4 Deriving PA Estimators in HJM Framework 86 3.5 Hedging MBS in HJM Framework 89 3.6 Hedging Performance Analysis 92 3.7 Conclusion 94 4. Hedging Credit Risk of MBS: A Random Coefficient Approach 95 4.1 Motivation 95 4.2 Literature Review 100 4.3 Introduction to Random Coefficient Model 104 4.4 Random Coefficient Model for Credit Spread Changes 107 4.5 Data Description 112 4.6 Results Analysis 115 4.6.1 Dependence of Credit Spread Sensitivities to State Variables 115 vi 4.6.2 Results by Rating and Maturity 123 4.6.3 Non-Constancy of Credit Spread Sensitivities 130 4.7 Conclusion and Future Work 135 Bibliography 136 vii LIST OF FIGURES 2.1 ?R(0,t) with Original Fourier series 25 2.2 ?R(0,t) with T 0 =10 modified Fourier series 25 2.3 Coefficients Estimation for Modified Fourier series 27 2.4 WAC as a function of Index 37 2.5 Gradient Estimator Comparison for ?d(t)/ ?? n 44 2.6 Gradient Estimator Comparison for ?c(t)/ ?? n 44 2.7 Gradient Estimator Comparison for ?PV(t)/ ?? n 45 2.8 95% Confidence Interval for dPV(t)/d? n 45 2.9 Gradient Estimator Comparison for ?d(t)/?? 46 2.10 Gradient Estimator Comparison for ?c(t)/?? 47 2.11 Gradient Estimator Comparison for ?PV(t)/?? 47 2.12 95% Confidence Interval for dPV(t)/d? 48 2.13 gamma estimators for 2 2 )( i td ?? ? , i=1, 2, 3, 4 49 2.14 gamma estimators for 2 2 )( i tCPR ?? ? , i=1, 2, 3, 4 49 2.15 gamma estimators for 2 2 )( i tCF ?? ? , i=1, 2, 3, 4 50 2.16 gamma estimators for 2 2 )( i tPV ?? ? , i=1, 2, 3, 4 50 2.17 Gradient Estimator Comparison for i tWAC ?? ? )( , i=1, 2, 3, 4 51 viii 2.18 Gradient Estimator Comparison for i tPV ?? ? )( , i=1, 2, 3, 4 52 2.19 Gradient Estimator Comparison for ?? ? )(tPV 52 2.20 Difference of FD/PA ?f(0, t)/ ?? n estimators 57 2.21 Fuction of xe -x 58 2.22 Duration vs. Products 62 2.23 The Impact of Modified Fourier Order 0 on FRM30 63 2.24 The Impact of Modified Fourier Order 1 on FRM30 64 2.25 The Impact of Modified Fourier Order 2 on FRM30 65 2.26 The Impact of Modified Fourier Order 3 on FRM30 66 2.27 The Impact of Modified Fourier Order 0 on ARM TSY 1 67 2.28 The Impact of Modified Fourier Order 1 on ARM TSY 1 68 2.29 The Impact of Modified Fourier Order 2 on ARM TSY 1 69 2.30 The Impact of Modified Fourier Order 3 on ARM TSY 1 70 2.31 10-Year T Rate, 1-Year T Rate, and mortgage rate 73 2.32 New ARM TSY 10 Durations 74 3.1 The first four principal components 82 3.2 Match monthly yield curve shift 83 3.3 Match annual yield curve shift 83 3.4 Mean Hedging Error of PCA vs. D&C 93 3.5 STD of Hedging Error: PCA vs. D&C 93 4.1 Credit Spread vs. Risk-free Rate 118 4.2 Credit Spread vs. Volatility 119 ix 4.3 Sensitivity to Volatility at different Leverage 120 4.4 Sensitivity to Volatility vs. Interest Rate 121 4.5 Sensitivity to Volatility vs. Maturity 122 4.6 Coefficient for ?r in RCR vs. Linear Model 130 4.7 Coefficient for ?vol in RCR vs. Linear Model 131 4.8 Coefficient for ?lev in RCR vs. Linear Model 131 4.9 Three-Month Treasury Rate from 1990 to 1997 133 4.10 VIX index from 1990 to 1997 133 x LIST OF TABLES 2.1 Product Specification for Mortgage Pricing 41 2.2 Comparison of PA/FD Duration 53 2.3 Comparison of Convexity Estimators 55 2.4 Comparison of Computing Costs 55 2.5 Durations of Different Products 60 3.1 Statistics for Principal Components 81 4.1 Comparison of three papers on credit spread regression 102 4.2 Comparison of RCR vs. linear model 116 4.3 Relationship between state variables and credit spread sensitivities 117 4.4 RCR coefficients for AA-AAA group 123 4.5 RCR coefficients for A group 124 4.6 RCR coefficients for BBB group 125 4.7 RCR coefficients for BB group 126 4.8 RCR coefficients for B and other group 127 4.9 Summary of RCR coefficients 128 xi LIST OF ABBREVIATIONS A(t) Amortization Factor at time t AGE(t) Aging Multiplier, a parameter to capture the aging effect in prepayment rate ARM Adjustable Rate Mortgage B(t) Balance at time t BM(t) Burnout Multiplier, a parameter to capture the burnout effect in prepayment rate C(t) Cash flow at time t CMO Collateralized Mortgage Obligation, a special type of MBS CPR Conditional Prepayment Rate, annualized prepayment rate CS(t) Credit Spread at maturity t D(t) Discounting Factor at time t f(0,t) Instantaneous forward rate starting from t observed at now FNMA Federal National Mortgage Association, also known as Fannie Mae FD Finite Difference FHLMC Federal Home Loan Mortgage Corporation, also known as Freddie Mac FRM Fixed Rate Mortgage GNMA Government National Mortgage Association, also known as Ginnie Mae GLS Generalized Least Square GSE Government-Sponsored Enterprise, mainly refers to Fannie Mae, Freddie Mac. xii H(t) Haircut at maturity t HJM Heath-Jarrow-Morton interest rate model, defined in Heath et al. [1992] IP(t) Interest Payment at time t lev t Leverage at time t LTV Loan to Value ratio, an 80 LTV loan means the loan amounts for 80% of the property value MBS Mortgage-Backed Securities MM(t) Monthly Multiplier, a parameter to capture the seasonal effect in prepayment rate MP(t) Mortgage Monthly Payment at time t OFHEO Office of Federal Housing Enterprise Oversight, a government agency under Department of Housing and Urban Development, regulator of Fannie Mae and Freddie Mac. OLS Ordinary Least Square PA Perturbation Analysis PCA Principal Components Analysis PDE Partial Differential Equation PMI Primary Mortgage Insurance PP(t) Principal Prepayment at time t PV(t) Present Value of Cash flow at time t R(0,t) Spot rate for maturity t observed at now r(t) Short rate at time t r 10 (t) 10-year rate at time t xiii RCR Random Coefficient Regression REO Real Estate Owned by the GSEs, in case borrower defaults RI(t) Refinance Incentive, a parameter to capture the refinance incentive effect in prepayment rate SMM Simple Monthly Mortality, monthly prepayment rate SP(t) Scheduled Principal Payment at time t TPP(t) Total Principal Payment at time t Vega The security price sensitivity to volatility WAC Weighted Average Coupon rate for MBS WAM Weighted Average Maturity for MBS w.r.t. with respect to 1 Chapter 1 Introduction Mortgage-backed securities (MBS) have become increasingly important fixed income instruments, both because of their volume and the role they play in fund investment and portfolio management. However, there has not been a very comprehensive set of risk indicators to measure and manage the risks involved with MBS. Hedging the interest rate and credit risk of MBS remains a complicated problem in the fixed income industry. This dissertation develops a set of risk measures for interest rate risk and credit risk, and then attempts to hedge the risks effectively using such risk measures. Specifically, the dissertation consists of three essays addressing the following problems: efficiently estimating these new measures of interest rate risk of MBS, hedging MBS with these new measures, and hedging the credit risk of MBS with advanced models for credit spread regression. The first essay is mainly positioned to answer the following research questions: ? How to measure the interest rate risk in a more comprehensive approach, rather than simply using the traditional duration 1 and convexity 2 ? 1 Duration is the first order derivative of the price of a fixed income security to interest rate, expressed as a percentage change, see Fabozzi [2001] for more details. 2 Convexity is the second order derivative of the price of a fixed income security to interest rate, expressed as a percentage change, see Fabozzi [2001] for more details. 2 ? How to efficiently estimate the risk measures if more factors are introduced into the measurement problem? In answering these two questions, based on the results over a broad spectrum of mortgage products, we propose a new mortgage product, which could be attractive to MBS investors and mortgage borrowers. The second essay tries to answer the following questions: ? What would be a realistic method to measure the term structure shift? ? How can we hedge MBS effectively with these measures, in a general interest rate model framework? We use Principal Components Analysis (PCA) method to extract the empirical volatility factors of term structure, which provides some justification for the form of possible term structure shifts proposed in the first essay. Then we use the Heath-Jarrow-Morton model to incorporate the factors in developing new risk measures and show that the hedging effectiveness is far better than traditional duration/convexity hedging. The third essay is related to credit risks MBS issuers incur when they purchase mortgage pool insurance from a third party, and attempts to answer the following questions: ? How to estimate the sensitivity of credit spread in a regression framework more effectively than a simple linear regression model? ? What implication does the model have on traditional structural models for credit spread? 3 We use a Random Coefficient Regression (RCR) model to build our regression model for credit spread changes. This model has explicit sensitivity measures dependent on state variables. We acquire much better explanatory power with this new model, without adding new independent state variables. Also our model supports the dependence of sensitivity of credit spread on state variables predicted by Merton?s structural model (Merton [1974]). 1.1 Efficient Sensitivity Analysis of Mortgage Backed Securities A mortgage-backed security (MBS) is a security collateralized by residential or commercial mortgage loans. An MBS is generally securitized, guaranteed and issued by three major MBS originating agencies: Ginnie Mae, Fannie Mae, and Freddie Mac. The cash flow of an MBS is generally the collected payment from the mortgage borrower, after the deduction of servicing and guaranty fees. However, the cash flows of an MBS are not as stable as that of a government or corporate coupon bond. Because the mortgage borrower has the prepayment option, mainly exercised when moving or refinancing, an MBS investor is actually writing a call option. Furthermore, the mortgage borrower also has the default option, which is likely to be exercised when the property value drops below the mortgage balance, and continuing mortgage payments would not be economically reasonable. In this case the guarantor is writing the borrower a put option, and the guarantor absorbs the cost. However, the borrower does not always exercise the options whenever it is financially optimal to do so, because there are always non- monetary factors associated with the home, like shelter, sense of stability, etc. And it is also very hard for the borrower to tell whether it is financially optimal to exercise these 4 options because of lack of complete and unbiased information, e.g., they may not be able to obtain an accurate home price, unless they are selling it. And there are also some other fixed/variable costs associated with these options, such as the commission paid to the real estate agent, the cost to initialize another loan, and the negative credit rating impact when the borrower defaults on a mortgage. All these factors contribute to the complexity of MBS cash flows. In practice, the cash flows are generally projected by complicated prepayment models, which are based on statistical estimation on large historical data sets. Because of the complicated behaviors of the MBS cash flow, due to the complex relationships with the underlying interest rate term structures, and path dependencies in prepayment behaviors, Monte Carlo simulation is generally the only applicable method to price MBS. Associated with the uncertainty of cash flows are different kinds of risks. Treasury bonds only bear interest rate risk, whereas non-callable corporate bonds carry interest rate and credit risk. MBS are further complicated by prepayment risk (resulting from both voluntary prepayment and default). Thus risk management is especially critical for portfolios with large holdings in MBS. Duration and convexity are the main risk measurements for fixed income portfolio mangers. Many practitioners use either the Macaulay duration, or modified duration (Kopprasch [1987]) to capture the MBS price sensitivity with respect to interest rate changes, but these duration measures assume a constant yield and known deterministic prepayment pattern, which is rarely the case in practice. So these two approaches to calculate duration can lead to serious errors when used in hedging. Golub [2001] proposed four different approaches to estimate the 5 duration: Percent of Price (POP), Option-Adjusted Duration (OAD), Implied Duration, and Coupon Curve Duration (CCD). The first two approaches apply parallel shifts in the yield curve, which is not a very realistic assumption. The latter two approaches require large numbers of previous or current accurate MBS prices that are comparable to the MBS whose duration is to be measured. This might not be practical for on-the-fly pricing and sensitivity analysis. Another drawback of these approaches is that they handle only duration and convexity, but not sensitivity to interest rate volatility. OAD method can estimate the vega (the price sensitivity to volatility) using a finite difference approach, which requires 3 simulations to estimate one gradient: the base, up and down cases. And non-parallel yield curve shifts require more parameters to characterize the shift. Thus, in the setting we consider in Chapter 2, to estimate the duration with respect to yield curve shift of 4 summed harmonic functions would require 9 (2n+1, n=4) simulations. To estimate vega requires 2 additional simulations. So estimating the duration and vega roughly increases the computational cost by a factor of 10. Calculating convexity would require 75 duration estimators to calculate 25 convexity estimators, increasing the simulation factor to 225. In other words, if one were to use 10,000 replications to estimate the MBS price, over 2.25 million simulations would be required to estimate the various sensitivities. Our work aims to decrease this computational burden dramatically. Most literature on MBS has concentrated on prepayment model estimations, although some of the recent work has focused on computational efficiency, e.g., dimensionality reduction via Brownian bridge ( Caflisch et al. [1997] ), and quasi-Monte Carlo (?kesson and Lehoczky [2000]). However, there is no work that we are aware of 6 that addresses efficient sensitivity analysis of MBS pricing and hedging. Related work in equities includes Fu and Hu [1995], Broadie and Glasserman [1996], Fu et al. [2000], [2001], and Wu and Fu [2001]. Perhaps the most relevant paper to our work is Glasserman [1999], which applied perturbation analysis (PA) method for caplet price sensitivity analysis. Yet most of these models involve only a single exercise decision with a one-time payoff, whereas an MBS is a pool of homogenous mortgages rather than an individual mortgage loan. So the cash flows exist until the maturity of the collateral, and they are highly path dependent, which makes sensitivity analysis of MBS more complicated. The other relevant body of research literature analyzes the duration of different mortgage products. We know that adjustable rate mortgage (ARM) products will have a different response from fixed rate mortgage (FRM) products, due to ARMs? coupon-reset plan and different prepayment function. In a series of papers, Kau et al.[1990,1992,1993] priced the ARMs and performed some sensitivity analysis. Chiang [1997] applied a simple simulation scheme to estimate the modified duration of ARMs. Stanton [1999] calculated the duration of different indexed ARMs via a scheme like Kau?s. However, most of these papers are based on solving models based on partial differential equation (PDE), using simplified assumptions that often miss essential features that can be captured by Monte Carlo simulation. The three major drawbacks of these models that make them impractical in the mortgage industry are the following: ? They assume borrowers exercise the prepayment option only when it is financially optimal to do so. This ignores the fact that people routinely prepay even in financially 7 adverse environments, e.g., house sales. Also seasoning and burnout effects are not considered. ? By solving the PDE, one can only obtain a set of present values of the MBS along the interest rate axis. By applying the finite difference method, duration of the MBS could be acquired. However, it provides no information about the discounting factor and cash flows along the time horizon. So you will have no information about how the interest shift affects different components of the present value. ? The PDE method generally uses one-factor interest rate model, which applies the same interest rate both for discounting and for the prepayment model, which ignores the difference between short-term and long-term interest rates. In the first essay, we apply perturbation analysis (PA) to estimate the sensitivities of MBS. Our work makes the following contribution to MBS literature: ? We decompose any interest rate term structure change into four Fourier-like series, which is based on Fourier cosine series, and can better measure the yield curve shift; ? We derive PA estimators for these Fourier-like factors, as well as interest rate volatility, which can largely save computation effort. In our example, we calculate 5 duration estimators and 16 convexity estimators in our simulation, which would require 155 simulations using a conventional simulation scheme. ? Based on our comprehensive analysis of the sensitivity measures we calculated for a full spectrum of mortgage product, we propose a new mortgage product, which can potentially benefit both the MBS investor and mortgage borrower. 8 This essay is organized in the following manner. Section 1 describes the problem setting. We then derive the framework for PA in a general setting in section 2, without restrictions to any specific interest rate model or prepayment model. Then we consider the well-known Hull-White interest rate model (Hull and White [1993]) and a common prepayment model to derive the corresponding PA sensitivities for FRM and ARM products in section 3, assuming the shock of interest rate term structure takes the form of a series of trigonometric polynomial functions. Section 4 presents numerical examples, in which we compare the performance of FD and PA estimators, indicating that the PA estimator is at least as good as the FD estimator, while the computation cost is reduced dramatically. Section 5 gives the insights from our simulation results. Section 6 gives conclusions. 9 1.2 Hedging MBS in HJM Framework There is a large body of literature on hedging with different interest risk measures, like first-order hedging with duration (Ilmanen [1992]), second order hedging with convexity (Kahn and Lochoff [1990], Lacey and Nawalkha [1993]), principal components hedging (Golub and Tilman [1997]), key rates hedging (Ho [1992]), level/slope/curvature hedging (Willner [1996]), etc. Yet there has not been a unifying effort in combining hedging the term structure together with hedging volatility factors. This essay tries to extract the empirical volatility factors from historical term structure data, via principal components analysis (PCA), and apply these factors in a HJM framework for pricing MBS, while deriving the risk measures for hedging MBS. It makes the following contribution in the MBS literature: ? The first paper to hedge MBS with PCA factors empirically extracted from historical interest rate data; ? Hedging efficiency is proved to increase significantly, compared with traditional duration/convexity hedging. This essay is organized in the following way. Section 1 gives the motivation for this research question. Section 2 describes the interest rate data set and PCA method we used to extract the volatility factors. Section 3 applies these factors in interest rate simulation within a HJM framework. Section 4 derives the PA estimators in the HJM framework. Section 5 gives the hedging algorithm for MBS, and Section 6 gives the performance analysis of this hedging method. Section 7 concludes the essay. 10 1.3 Hedging Credit Risk of MBS: A Random Coefficient Approach In order to hedge the credit risk of MBS, the MBS issuer sometimes needs to purchase pool insurance from a third party, beyond the protection of mortgage collateral, and primary mortgage insurance. In this case, it is important to model the credit risk of the third party. Recently there has been increased interest in some research papers to use regression method to determine what factors affect credit spread. Most of the papers, which use simple linear regression, found that variables in structural models lack explanatory power in such regression. We argue that the problem partially results from non-constant credit spread sensitivities to state variables. We try to overcome the problem by proposing a Random Coefficient Regression (RCR) model. We collected data from multiple database, and constructed our data set. Our regression results show that our assumption of non-constancy of credit spread sensitivities is correct. As a result of improved regression, we improved adjusted R 2 to 28%, compared with 8% adjusted R 2 for a simple linear regression approach, using the same set of independent variables. Another important result of our RCR model is that it validated the relationship between credit spread sensitivities and state variables, which has been predicted by Merton?s model. This essay makes the following research contributions to the finance literature: ? The first paper to use the RCR method on credit spread data; 11 ? The first paper to explicitly build a dependence relationship between credit spread sensitivity and state variables; ? The first paper to empirically validate the dependence relationship between credit spread sensitivity and state variables predicted by a structural model, such as Merton?s model. This essay is organized in the following way. Section 1 gives the motivation for this research question. Section 2 describes several previous papers on this topic. Section 3 gives a brief introduction to the Random Coefficient Regression model. Section 4 applies this model to changes of credit spread. Section 5 gives the data description used in the regression, and Section 6 gives the results analysis of this regression method. Section 7 concludes the essay. 12 Chapter 2 Efficient Sensitivity Analysis of MBS 2.1 Problem Setting Generally the price of any security can be written as the net present value (NPV) of its discounted cash flows. Specifying the price of an MBS (here we consider only the pass-through MBS 1 ) is as follows: ? ? ? ? ? ? = ? ? ? ? ? ? == ?? == M t M t tctdEtPVEVEP 00 )()()(][ , (2.1) where P is the price of the MBS, V is the value of the MBS, which is a random variable, dependent on the realization of the economic scenario, PV(t) is the present value for cash flow at time t, d(t) is the discounting factor at time t, c(t) is the cash flow at time t, M is the maturity of the MBS. 1 A pass-through MBS is an MBS that passes through the principal and interest payments collected from a mortgage pool, minus the guaranty fee and servicing fee, to the MBS investor directly. This is in contrast to Collaterized Mortgage Obligations (CMOs), which have multiple tranches and pay the principal payments according to the seniorities of tranches. In this essay, we assume that mortgages in the MBS pool are homogenous. 13 Monte Carlo simulation is used to generate cash flows on many paths. By the strong law of large numbers, we have the following: [] ? = ?? = N i iN V N VE 1 1 lim , (2.) where V i is the value calculated out in path i. The calculation of d(t) is found from the short-term (risk-free) interest rate process, ? ? ? = ? = ??=??=?= 1 0 1 0 }])([exp{))(exp(),1()2,1()1,0()( t i t i tirtirttdddtd , (2.3) where d(i, i+1) is the discounting factor for the end of period i+1 at the end of period i; r(i) is the short term rate used to generate d(i, i+1), observed at the end of period i; ?t is the time step in simulation, generally monthly, i.e. ?t= 1 month. An interest rate model is used to generate the short term-rate r(i); then d(t) is instantly available when the short-term rate path is generated. The difficult part is to generate c(t), the path dependent cash flow of MBS for month t, which is observed at the end of month t. From chapter 19 of Fabozzi [1993], we have the following formula for c(t): );()()( );()()( );()()()()( tPPtSPtTPP tIPtSPtMP tIPtTPPtPPtMPtc += += +=+= (2.4) where MP(t): Scheduled Mortgage Payment for month t; TPP(t): Total Principal Payment for month t; 14 IP(t): Interest Payment for month t; SP(t): Scheduled Principal Payment for month t; PP(t): Principal Prepayment for month t. These quantities are calculated as follows: ;)(11)( );()1()( ));()1()(()( ; 12 )1()( ; )12/1(1 12/ )1()( 12 tCPRtSMM tTPPtBtB tSPtBtSMMtPP WAC tBtIP WAC WAC tBtMP tWAM ??= ??= ??= ?= ? ? ? ? ? ? ? ? +? ?= +? (2.5) B(t): The principal balance of MBS at end of month t; WAC 2 : Weighted Average Coupon rate for MBS; WAM 3 : Weighted Average Maturity for MBS; SMM(t): Single Monthly Mortality for month t, observed at the end of month t; CPR(t): Conditional Prepayment Rate for month t, observed at the end of month t. In Monte Carlo simulation, along the sample path, CPR(t) is the primary variable to be simulated. Everything else can be calculated out once CPR(t) is known. Different prepayment models offer different CPR(t), and it is not our goal to derive a new 2 WAC is the weighted average mortgage rate for a mortgage pool, weighted by the balance of each mortgage. 3 WAM is the weighted average maturity in month for a mortgage pool, weighted by the balance of each mortgage. 15 prepayment model or compare existing prepayment models. Instead, our concern is, given a prepayment model, how can we efficiently estimate the price sensitivities of MBS against parameters of interest? Generally different prepayment models will lead to different sensitivity estimates, so it is at the user?s discretion to choose an appropriate prepayment function, as our method for calculating the ?Greeks? is universally applicable. 2.2 Derivation of General PA Estimators If P, the price of the MBS, is a continuous function of the parameter of interest, say ?, and assuming the interchange of expectation and differentiation is permissible 4 , we have the following PA estimator by differentiating both sides of (2.1): ).,( ),( ),( ),()),(( , ),( ),( )]([)( 1 1 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? td tc tc td d tPVd d tdPV E d tPVd E d VdE d dP M t M t ? ? + ? ? = ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? ? ? == ? ? = = (2.6) Now we have reduced the original problem from estimating the gradient of a sum function to estimating the sum of a bunch of gradients. Actually now we only need to estimate two gradient estimators, ? ? ? ? ),(tc and ? ? ? ? ),(td , at each time step. 4 Strictly speaking, sample pathwise continuity of price function with respect to ? will result in the interchange being valid. 16 2.2.1 Gradient Estimator for Cash Flow We first derive ? ? ? ? ),(tc . To simplify notation, we write c(t) for c(t, ?). A simplified expression for c(t) is derived from (2.4) and (2.5) as follows: {},)()](1)[()1( )() 12 1)(1()](1[ )12/1(1 12/ )1( )() 12 1)(1())(1)(( )()]}()([)1({)( )()]()1([)()()()( tSMMgtSMMtAtB tSMM WAC tBtSMM WAC WAC tB tSMM WAC tBtSMMtMP tSMMtIPtMPtBtMP tSMMtSPtBtMPtPPtMPtc tWAM +??= +?+? +? ?= +?+?= ???+= ??+=+= +? (2.7) where ). 12 1( , )12/1(1 12/ )( WAC g WAC WAC tA tWAM += +? = +? (2.8) Then we can derive the gradient for c(t), if WAC and t are independent 5 of ?: {} ])()[1( )( )()](1)[( )1()( gtAtB tSMM tSMMgtSMMtA tBtc +?? ? ? ++? ? ?? = ? ? ??? . (2.9) This leads to recursive equations for calculation of the above gradient estimator from (2.5) and (3.2): . )()1()( );()1()() 12 1)(1()()1()( );1( 12 )()()()( ??? ? ? ? ? ?? = ? ? ??=?+?=??= ??=?= tc g tBtB tcgtBtc WAC tBtTPPtBtB tB WAC tctIPtctTPP (2.10) 5 A fixed Rate Mortgage (FRM) would satisfy this assumption; however an Adjustable Rate Mortgage (ARM) will not, so we derive the gradient estimator for ARMs later in section 4. 17 Assuming we know that the initial balance is not dependent as ?; we have the initial conditions: ).)1()(0( )1()1( ;0 )0( gAB SMMc B +? ? ? = ? ? = ? ? ?? ? (2.1) This leads to the following , )1()0()1( ??? ? ? ? ? ? = ? ? c g BB (2.12) {}),)2()(1( )2( )2()2(1)(2( )1()2( gAB SMM SMMgSMMA Bc +? ? ? ++? ? ? = ? ? ??? (2.13) Mt tc ,...,2,1, )( = ? ? ? . Thus the problem of calculating the gradient estimator of cash flow c(t) is reduced to calculating: .,...,1, )( Mt tSMM = ? ? ? Since ,)(11)( 12 tCPRtSMM ??= we have . )( ))(1( 12 1)( 12 11 ?? ? ? ?= ? ? ? tCPR tCPR tSMM (2.14) As discussed earlier, generally CPR(t) is given in the form of a prepayment function, and there are four main types of prepayment functions (Fabozzi [2000]): 1. Arctangent Model: (An example from the Office of Thrift Supervision (OTS).) 18 )) )( 089.1(9518.5arctan(1389.02406.0)( 10 tr WAC tCPR ??= . (2.15) 2. CPR(S,A,B,M) Model: );()()()()( tBMtMMtAGEtRItCPR = (2.16) where RI(t) is refinancing incentive; AGE(t) is the seasoning multiplier; MM(t) is the monthly multiplier, which is constant for a certain month; BM(t) is the burnout multiplier. 3. Prepayment models incorporating macroeconomic factors, i.e., the health of economics, housing market activity, etc. 4. Prepayment models for individual mortgages. For the last two types of prepayment models, we do not have any explicitly stated functional forms, mainly because they are proprietary models in the mortgage industry. But since our approach is general for any type of prepayment function, we can derive the derivatives once we are given an explicit form for the prepayment function. We would like to make one claim here: the CPR(t) model we mentioned and we are going to use in Chapter 2 and Chapter 3 includes both voluntary prepayment (refinance, house turnover, and cash out) and involuntary prepayment (default). Because default only makes about 1% of total prepayment, and generally MBS issuer will guarantee the principal payment to the investor, in case borrower defaults, it is a reasonable not to model default separately. However, when analyzing MBS backed by high default loans, such as subprime mortgages, it is desirable to model voluntary prepayment and default separately. 19 2.2.2 Gradient Estimator for Discounting Factor We have derived the gradient estimator of cash flow with respect to parameter ?. Next, we derive the gradient estimator of the discounting factor d(t). We know that the discounting factor takes the following form from section 2, when the option adjusted spread (OAS) is not considered. For simplification, we write d(t) as for d(t, ?): }])([exp{)( 1 0 tirtd t i ??= ? ? = . (2.17) Differentiating with respect to ?: .) )( ()() )( (}])([exp{ )( 1 0 1 0 1 0 t ir tdt ir tir td t i t i t i ? ? ? ?=? ? ? ???= ? ? ??? ? = ? = ? = ??? (2.18) From the gradient estimators for cash flow and discounting factor, we can easily get the gradient estimator of PV(t): ),( ),( ),( ),()),(( ? ? ? ? ? ? ? ? td tc tc td d tPVd ? ? + ? ? = . (2.19) The last step would be to apply a specific prepayment model and interest rate model to arrive at the actual implemented gradient estimators. To illustrate the procedure, we carry out this exercise in its entirety for one setting in the following section. 20 2.3 Applying the Gradients We choose our interest model to be the one-factor Hull-White (Hull and White [1990]) model, for its simplicity and easy calibration to market term structure. For the prepayment model, we consider a CPR(S,A,B,M) model. 2.3.1 Interest Model Setup In this section, we briefly discuss the model and the simulation scheme. In the one-factor Hull-White interest rate model, the underlying process for the short-term rate r(t) is given by ),())()(()( tdBdttarttdr ?? +?= (2.0) where B(t): a standard Brownian motion; a: mean reverting speed, constant; ?: standard deviation, constant; ?(t): chosen to fit the initial term structure, which is determined by ),1( 2 ),0( ),0( )( 2 2 at e a taf t tf t ? ?++ ? ? = ? ? (2.1) f(0,t): the instantaneous forward rate, which is determined by ,),0( 1 ),0( 0 ? = t duuf t tR (2.2) Differentiating both sides, with respect to t, we have ),0( ),0( ),0( tR t tR ttf + ? ? = , (2.3) where R(0,t): the continuous compounding interest rate from now to time t, i.e. the term structure. 21 In order to simplify the simulation process, the model can be re-parameterized from its original to the following: ;0)0(),()()()( =+?= xtdBdttxtatdx ? (2.4) x(t) is determined by 2 2 )1( 2 ),0()()()( at e a tftxtrta ? ?+=?= ? . (2.5) The process x(t) is called an Ornstein-Uhlenbeck process, and its solution is given by ? ? = t auat udBeetx 0 )()( ? , (2.6) which is a Gaussian Markov process, and can also be represented as ) 2 1 ()( 2 a e Wetx at at ? = ? ? , (2.7) where {W(t), t?0} is also a Brownian motion. In this case, the interest rate r(t) can be represented in the following form: ))()(()( )(th WtgtaFtr += , (2.8) where a, g: R + ? R are continuous functions, and the functions F:R ? R and h: R + ? R are strictly increasing and continuous. From above we can see that . 2 1 )( ;)( ;)1( 2 ),0()( ;)( 2 2 2 2 a e th etg e a tfta xxF at at at ? = = ?+= = ? ? ? ? (2.9) To simulate r(t) given by above, we will first simulate )( )()( th Wtgtx = , 22 which is a Gaussian Markov process, and then compute the short-term interest rate by ))()(()( txtaFtr += . For calculating the price of MBS, the short-term rate is not sufficient; the long- term rate process is also required, especially the 10-year Treasury rate, which is a deterministic function of r(t) in the Hull-White model. Generally this is the case for short-term rate models, but not true for more complicated interest rate models, e.g., the HJM (Heath, Jarrow and Morton [1992]) model and the LIBOR forward rate model (Jamshidian[1997]). The long-term rate R(t,T) is calculated from the following, : ).1()( 4 ),0(ln ),( ),0( ),0( ln),(ln ; 1 ),( ;),(),( 22 3 2 )( )(),())(,( ??? ? ? ?= ? = == ?? ?? ??? atataT tTa trTtBtTTtR eee at tP TtB tP TP TtA a e TtB eTtAeTtP ? (2.30) P(t,T) is the zero coupon bond price at time t, with face value $1, matured at T. Thus we can derive the R(t,T) as following: )( )(),(),(ln ),( tT trTtBTtA TtR ? ? ?= . (2.31) The standard (forward) path generation method for generating x(t) is given by ,))(())(()()( )( )( ))](())(()[())(()( )( )( )( 111 1 11 1 1 +++ + ++ + + ?+= ?+= iiiii i i iiiii i i i zthWthWtgtx tg tg thWthWtgthWtg tg tg tx (2.32) where {z i } is a series of independent standard normal random variables. In the special case where x(t) is from the Hull-White model, we have 1 2 1 2 1 )()( + ?? ?? + ? += i ta i ta i z a e txetx i i ? , (2.33) 23 where ?t i =t i+1 - t i . 2.3.2 Trigonometric Polynomial Shocks There are multiple factors in the interest rate model that can change and affect the cash flows and discounting factor along the simulation path. The major changes could be the initial term structure R(0,t) and the volatility ?. The most common assumption for term structure change is a parallel shift on all maturities. However, this is often not an adequate model for the real world, where a shift in the term structure can take any shape. For example, short-term rates and long-term rates may change in opposite directions rather than in parallel. We consider a Fourier series decomposition of the term structure shift. Our domain of concern is interest rates from time 0 to 30 years, since most mortgages are amortized in a 30-year term. So for example, we could assume the shift of term structure takes the following form: ? ? = ?=? 0 ), 30 cos(),0( n n tn tR ? (2.34) where ? n is the magnitude for the n th Fourier function. Figure 4.1 depicts the first four trigonometric polynomial series. (n=0,1,2,3), which is all that we will consider in our model. When n=0, the shift is just like a parallel shift in term structure. When n=1, the short-term and long-term rates move in opposite directions. When n=2, the short-term and long-term rates move in the same direction, while the middle-term rate moves in the opposite direction. Thus we decompose any shift in the term structure into the Fourier functions by Fourier transform. If we have previously calculated the gradients with respect to the magnitude of each trigonometric polynomial function, we can apply these 24 gradients and get the corresponding changes in the cash flows and discounting factors, and hence the change in MBS prices. The Fourier series have a serious drawback: they treat short-term rates the same as long-term rates. However, from experience, we know that the short-term rates generally change more frequency than long-term rates. So we would like to change the shape of the trigonometric polynomial function, which will concentrate more on the short-term rates, and keep the long-term rates relatively stable. The modified Fourier function that we adopt takes the following form: ? ? = ? ??=? 0 / ))1(cos(),0( 0 n Tt n entR ? , (2.35) where T 0 is a user-specified parameter of the modified Fourier shifts. The smaller T 0 is, the more likely short rates and long rates are going to act differently. See Figure 2.2 for the modified Fourier functions, where T 0 =10. Comparing Figures 2.1 and 2.2, the modified Fourier series concentrate more on the changes with maturities less than T 0, which is both desirable and easier for analytical purposes. For a Fourier cosine series that has the following functional form: ? ? = += 1 0 ) 2 cos( 2 )( n n T tn a a tf ? , (2.36) the coefficients are given by a Fourier cosine transform: ? == 2/ 0 210 ,) 2 cos()( 4 T n ,......,,ndt T tn tf T a ? (2.37) 25 Figure 2.1 ?R(0,t) with Original Fourier series Figure 2.2 ?R(0,t) with T 0 =10 modified Fourier series 26 For our modified Fourier series, perform the following change of variables: )1( 30 ' 0 / Tt e t ? ?= and substitute into the expression of ?R(0,t) to get ? ? = ?=? 0 ) 60 '2 cos()',0( n n tn tR ? , (2.38) which is a standard Fourier cosine series, and we can use a Fourier transform to estimate the coefficients. In computer simulation, t is a vector of real time points, evenly distributed with sample function value ?R(0,t), and t? is the mapped time point in a new time scale, which is not evenly distributed, with the same sample function value ?R(0,t?). However, in order to utilize the discrete cosine transform function provided in mathematical libraries, we need to resample ?R(0,t?) at even time intervals. This is carried out by interpolating the function of ?R(0,t?) on the t time scale. Figure 2.3 shows a sample of ?R(0,t), ?R(0,t?) re-sampled on t, the coefficients estimated on the re- sampled ?R(0,t?), and the reconstructed Fourier series of ?R(0,t). From Figure 2.3, if we look at the upper left and lower right sub-figures, we can see that the reconstructed term structure matches the original sample very well, which validates our method for estimating the coefficients of the modified Fourier series. 27 Figure 2.3 Coefficients Estimation for Modified Fourier series 2.3.3 Derivation of Gradients with respect to Modified Fourier Functions Our major task in this section is to derive the gradient estimator with respect to to specific parameters in the interest rate model and prepayment model. Specifically, for the former, we are interested in the parameters of the modified Fourier functions (? n, n=0, 1, 2, 3). First we derive the discounting factor gradient estimator. From (3.12), we know that in order to derive ?? ? )(td , we must first derive ?? ? )(ir , i=0,?, t-1. Let us recall that in section 4.1, we have the following simulation scheme for short term rate r(t): )()()( txtatr += . 28 So ??? ? ? + ? ? = ? ? )()()( txtatr , (2.39) where ?? ? )(ta and ?? ? )(tx are determined as the following in Hull-White model: .0 )( , ),0()( = ? ? ? ? = ? ? ? ?? tx tfta (2.40) We also know the relationship between f(0,t) and R(0,t) from (2.23), so ?? ? ),0( tf can be derived as: . ),0(),0( ),0(),0(),0( ),0( ),0( ),0( 2 2 ?? ??? ??? ? ? + ?? ? = ? ? + ?? ? + ? ? ? ? = ? ? + ? ? ? ? ? ? ? ? ? ? = ? ? tR t tR t tR t tR t t tRt tRt tR t tf (2.41) Considering the changes in R(0,t) which takes the form as in (2.38), we can get the derivatives of R(0,t) (? taken to be ? n ): . )( ))1(sin( )( ))1(sin( ),0( )),1(cos( ),0( 0 / / 0 / / 2 / 0 0 0 0 0 T en en T en en t tR en tR Tt Tt Tt Tt n Tt n ? ? ? ? ? ?= ? ? ??= ??? ? ?= ?? ? ? ? ? ? ? (2.42) We can get the derivatives of r(i): )).1(cos( )( ))1(sin( ),0()( 0 0 0 / 0 / / Tt Tt Tt nn en T en ent tfir ? ? ? ?+??= ?? ? = ?? ? ? ? ? (2.43) And gradient estimator for discounting factor is also obtained, applying (2.18). 29 Next, we are going to derive the cash flow gradient estimator with respect to ? n . From our derivation in section 3, we know that in order to get ?? ? )(tc , we need to derive ?? ? )(tCPR first. We use the second type of prepayment function, among the four described in section 3. An example for this type of prepayment model is available from the sample code at the Numerix homepage http://www.numerix.com. );()()()()( tBMtMMtAGEtRItCPR = (2.4) where rate. mortgage prevailing ith the w correlatedhighly is which t,period of end at the observed rate,year 10 theis )( ; )0( )1( 7.03.0)( December;in ending January, from starting 0.98], 1.23, 1.22, 1.18, 1.1, 0.98, 0.92, 0.98, 0.95, 0.74, 0.76, [0.94,)( ); 30 ,1min()( )));1((430571.8arctan(14.028.0)( 10 10 tr B tB tBM tMM t tAGE trWACtRI ? += = = ??+?+= From the formulas, only RI(t) and BM(t) depend on ?, when ? is not time t. Thus we have the following formula for ?? ? )(tCPR : ; )( )()()()()()( )()( ??? ? ? + ? ? = ? ? tBM tMMtAGEtRItBMtMMtAGE tRItCPR (2.45) where . )0( 1)1( 7.0 )( ; )1( )))1((430571.8(1 )430( 14.0 )( 10 2 10 B tBtBM tr trWAC tRI ?? ?? ? ?? = ? ? ? ?? ??+?+ ? = ? ? (2.46) 30 ?? ? )(tB is available, when ?? ? )(tc is calculated out, so the problem is reduced to calculating ?? ? )( 10 tr . In the one-factor Hull-White framework, as we have discussed in section 2.3.1, the long-term rate is a deterministic function of r(t), so substituting T=t+10 for (2.30), we have ).1()( 4 ),0( 1 ),0()10)(10,0( )1()( 4 ),0(ln )10,(),0(ln)10,0(ln )1()( 4 ),0(ln )10,( ),0( )10,0( ln)10,(ln ; 11 )10,( ;)10,()10,( 22)10( 3 210 22)10( 3 2 22)10( 3 2 10)10( )()10,()10)(10,( ??? ? ++++?= ??? ? ? +??+= ??? ? ? +? + =+ ? = ? =+ +==+ ?+? ? ?+? ?+? ??+? +??++? atatta a atatta atatta atta trttBttttR eee a tR a e ttRttR eee at tP ttBtPtP eee at tP ttB tP tP ttA a e a e ttB ettAettP ? ? ? (2.47) Since , 10 )( 1 )1()( 4 ),0( 1 ),0()10)(10,0( 10 )()10,()10,(ln )10,()( 10 22)10( 3 210 10 tr a e eee a tR a e ttRttR trttBttA ttRtr a atatta a ? ?+? ? ? ???? ? ++++? ?= +?+ ?=+= ? (2.48) ?? ? )( 10 tr takes the following form, when ? is independent of ? and t: 10 )(1),0( ) 1 ( )10,0( )10( )( 1010 10 ??? ? ? ?? ? ? ?? ++ ? +? +? ?= ? ? ?? tr a etR a e t tR t tr aa . (2.49) Thus we have derived ?? ? )( 10 tr as a function of ?? ? )(tR and ?? ? )(tr derived earlier. 31 2.3.4 Derivation of Gradients with respect to Volatility: Vega The derivation is straightforward as in section 2.3.3; all we need to do is to substitute ? with ?, instead of ? n . In order to get ?? ? )(td , we must first derive ?? ? )(ir . Following the same logic in (2.40), we can get the vega of r(t): ). 2 1 ()1( )( so), 2 1 ( )( ,)1( )( 2 2 2 2 2 2 a e Wee a tr a e We tx e a ta at atat at at at ? +?= ? ? ? = ? ? ?= ? ? ?? ? ? ? ? ? ? ? (2.50) And vega of d(t) would be: .) )( ()( )( 1 0 t ir td td t i ? ? ? ?= ? ? ? ? = ?? (2.51) Now we derive ?? ? )(tc , which would require us to derive ?? ? )(tCPR first, which has the same form as in (2.45), while ?? ? )( 10 tr has the form of: . 10 )(1 )1()( 2 10 )(1 )1()( 2 ),0( ) 1 ( )10,0( )10( )( 10 22)10( 3 10 22)10( 3 10 10 ? ? ? ? ?? ? ? ?? +?? = ? ?? ???? ? ?? ++ ? +? +? ?= ? ? ? ?+? ? ?+? ? tr a e eee a tr a e eee a tR a e t tR t tr a atatta a atatta a (2.52) 2.3.5 Derivation of Second Order Gradients: Gamma Another gradient that interests risk mangers is convexity, or the gamma of MBS, which is the second order derivative of price against term structure shifts. Now we derive an estimator for the gamma. 32 In order to calculate the partial second order derivatives (Hessian matrix), we take ? to be the vector, ?=[? 1 ? 2 ? 3 ? 4 ?]?. Differentiating (2.1) 1 , we get .)( )( ' )()( ' )()( )( )()(][ ,)( )( )( )()(][ 0 2 2 2 2 0 2 2 2 2 2 2 00 ? ? ? ? ? ? ? ? + ? ? ? ? ? + ? ? ? ? ? + ? ? = ? ? ? ? ? ? ? ? = ? ? = ? ? ? ? ? ? ? ? ? ? + ? ? = ? ? ? ? ? ? ? ? = ? ? = ? ? ?? ?? == == M t M t M t M t td tctdtctctd tc td E tPV E VEP td tc tc td E tPV E VEP ????????? ????? (2.53) where ]' [ 4321 ?? ? ? ?? ? ?? ? ?? ? ?? ? = ? ? PPPPPP , and 2 2 ?? ? P is a 5-by-5 matrix, whose (i, j) th element is determined by ji P ?? ?? ? 2 , where ? I and ? I are the i th and j th elements of ?, respectively. The same notation will be used for gradients of other variables, i.e. c(t), d(t), r(t), etc. Since we have calculated ?? ? )(tc and ?? ? )(td in previous sections, now the problem is reduced to estimate 2 2 )( ?? ? tc and 2 2 )( ?? ? td . So we first derive the gamma for the discounting factor d(t). Differentiating (2.18), we get t irtd t ir td td t i t i ?? ? ? ?? ? ? +?? ? ? ??= ? ? ?? ? = ? = )' )( ( )( ) )( ()( )( 1 0 1 0 2 2 2 2 ???? (2.54) Once we have 2 2 )( ?? ? ir , the gamma of d(t) is easily calculated. Now we derive the gamma for cash flow c(t). From (2.9), we can derive the following gamma equation: 1 Again, we need the first order derivative to be pathwise continous to make the interchange of expectation and differentiation permissible. 33 {} ].)()[1( )( ])(][' )1()( ' )()1( [ )()](1)[( )1()( 2 2 2 2 2 2 gtAtB tSMM gtA tBtSMMtSMMtB tSMMgtSMMtA tBtc +?? ? ? + +? ? ?? ? ? ? + ? ? ? ? ?? + +? ? ?? = ? ? ? ???? ?? (2.55) And from (3.5), we can get the gamma of B(t): 2 2 2 2 2 2 )()1()( ??? ? ? ? ? ?? = ? ? tc g tBtB . (2.56) Now we calculate gamma of SMM(t): .,...,1, )( 2 2 Mt tSMM = ? ? ? As we know from (3.9), we have . )( ))(1( 12 1 )' )()( ())(1( 144 11)( 2 2 12 11 12 23 2 2 ???? ? ? ?+ ? ? ? ? ? ?= ? ? ?? tCPR tCPR tCPRtCPR tCPR tSMM (2.57) ?? ? )(tCPR and 2 2 )( ?? ? tCPR will be prepayment model specific. For discounting factors, if we choose the Hull-White one factor model, we have the following: .5,1, )()( ;]' )( )( )( )( )( [ )( 2 2 2 4321 ?? ? ? ? ? ? ? ? ? ?? ? = ? ? ? ? ?? ? ?? ? ?? ? ?? ? = ? ? ji irir iriririririr ji ??? ?? (2.58) From and (2.43) and (2.49), we can derive the following: 34 . )1()( ;0 )( ;0 )( 2 2 2 2 2 2 a eir ir ir tai i ji ?? ? = ? ? = ??? ? = ???? ? ? ? (2.59) And the gamma of d(t) would be )(/' )()( )' )( ( )( )' )( ( )( ) )( ()( )( 1 0 1 0 1 0 2 2 2 2 td tdtd t irtd t irtd t ir td td t i t i t i ?? ?? ???? ? ? ? ? ? = ?? ? ? ?? ? ? = ?? ? ? ?? ? ? +?? ? ? ??= ? ? ? ?? ? = ? = ? = (2.60) For cash flows, based on the equations (2.45) and (2.48) in the CPR(S, A, B, M) model, we have: , )0( 1 * )1( *7.0 )( ; )1( )))1((430571.8(1 5/301 )' )1()1( )( )))1((430571.8(1 5/301 ( )( ]; )( )(' )()( ' )()( )( )( )[()( )( 2 2 2 2 2 2 10 2 10 1010 2 1010 2 2 2 2 2 2 2 2 B tBtBM tr trWAC trtr trWACr tRI tBM tRI tRItBM tBMtRI tBM tRI tMMtAGE tCPR ?? ? ??? ??? ???? ? ?? = ? ? ? ?? ??+?+ + ? ?? ? ? ?? ??+?+? ? = ? ? ? ? + ? ? ? ? ? + ? ? ? ? ? +? ? ? = ? ? (2.61) where we know from (2.58) that 35 . 10 )(1 )( 0; )10,0( ;0 ),0( ; 10 )(1),0( ) 1 ( )10,0( )10( )( 210 2 10 2 2 2102102 2 10 ji a ji ji ji ji a ji a ji ji tr a e tr tR tR tr a etR a e t tR t tr ?? ?? ?? ?? ?????? ?? ?? ?? = ?? ? = ?? +? = ?? ? ?? ?? ? ?? ?? ++ ?? +? +? ?= ?? ? ? ?? (2.62) Finally, the gamma of price P given by equation (2.53) can be obtained from equations (2.60), (2.61), and (2.62). 2.3.6 Derivation of ARM PA estimators In this section, we derive PA estimators for ARMs. We know FRMs only have two sources of uncertainty: ? Short-term rate r(t), which affects the discounting factor d(t), and ? Long-term rate r 10 (t), which determines the prepayment rate CPR(t), and hence determines the cash flow C(t). ARMs introduce one more source of uncertainty, the coupon rate WAC(t), which affects both the amortization schedule and the prepayment rate CPR(t), and then affects the cash flow C(t). Coupon rate is determined by many factors: ? The index rate. WAC resets to the index rate plus the margin periodically. ? Margin. The spread between the WAC and the index rate. ? Adjustment period. For fixed period (FP) ARMs, the first adjustment period is different from subsequent adjustment period. 36 ? Period Cap/Floor. The maximum amount the WAC could increase/decrease from previous period. ? Lifetime Cap/Floor. The maximum/minimum coupon rate over the lifetime of the mortgage. In order to derive the PA gradient estimator of C(t) for ARM, we first need to derive the PA gradient estimator for Index(t) and WAC(t). The most commonly used index rate is the 1-year Treasury rate. In the Hull-White model, it is an explicit function of short-term rate r(t) and the term structure R(0, t). As we have derived the function form of r 10 (t), we can derive the r lag (t) for any lag: (in this case, lag=1) . )( 1 )1()( 4 1 ),0( 1 ),0())(,0( )(),(),(ln ),()( * 22)( 3 * lag tr a e eee a tR a e ttRlagtlagtR lag trlagttBlagttA lagttRtr alag atatlagta alag lag ? ?+? ? ? ???? ? ??++? ?= +?+ ?=+= (2.63) Thus we have the PA gradient estimator of ?? ? )(tIndex in following form: lag tr a etR a e t lagtR lagt tr tIndex alagalag lag ??? ?? ? ?? ? ? ?? +? ? +? +? ?= ? ? = ? ? ?? )(1),0( ) 1 ( ),0( )( )( )( ** . (2.64) The hard part is to get the ?? ? )(tWAC from ?? ? )(tIndex , because of the complicated rules to determine WAC(t), based on all the factors mentioned above. Given WAC(t-1), Index(t), 37 Margin, Period_Cap 2 , Period_Floor, Life_Cap, Life_Floor, WAC(t) is determined as follows: ? ? ? ? ? ?+ +? <+<+ = += = ?= ;Margin if ;Margin if ;Margin ifMargin 1max 1min Otherwise, moment; adjustmentan not is if ),1()( CapEffective_Index(t)Cap, Effective_ Index(t)FloorEffective_Floor, Effective_ CapEffective_Index(t)FloorEffective_, Index(t) WAC(t) ); Period_Cap) WAC(t-(Life_Cap,CapEffective_ loor);)-Period_Fr, WAC(t-(Life_FlooFloorEffective_ ttWACtWAC (2.65) Figure 2.4 shows the relationship of WAC with Index. Figure 2.4 WAC as a function of Index 2 Life_Cap/Life_Floor are absolute numbers, while Period_Cap/Period_Floor are relative. 38 Then we can derive the ?? ? )(tWAC as following: ? ? ? = ? ? ? ? ? ? ? ? ? ?+< ? ?? +?> ? ?? <+< ? ? = ? ? += = ? ?? = ? ? false. iscondition when 0, true;iscondition when ,1 }{ where ;Margin if}__{* )1( ;Margin if}__{* )1( ;Margin if 1max 1min Otherwise, moment; adjustmentan not is t if , )1()( conditionI CapEffective_Index(t), CapLifeCapEffectiveI tWAC Index(t)FloorEffective_, FloorLifeFloorEffectiveI tWAC CapEffective_Index(t)FloorEffective_, Index(t) WAC(t) ); Period_Cap) WAC(t-(Life_Cap,CapEffective_ loor);)-Period_Fr, WAC(t-(Life_FlooFloorEffective_ tWACtWAC ? ? ? ? ?? (2.6) Note that the gradient is 0, when it is bounded by lifetime cap or floor, because a perturbation would not change the WAC(t). Next, we need to derive ?? ? )(tCPR for ARM, assuming ARM borrowers have the same prepayment behavoir as FRM borrowers (which is not necessarily true, but it does not affect our analysis), so we are facing the same prepayment function as FRM30 as in (2.45). ?? ? )(tCPR will be affected because of the uncertainty of WAC(t). 39 . )0( 1)1( 7.0 )( ]; )1()( [ )))1()((430571.8(1 430 14.0 )( ; )( )()()()()()( )()( 10 2 10 B tBtBM trtWAC trtWAC tRI tBM tMMtAGEtRItBMtMMtAGE tRItCPR ?? ??? ??? ? ?? = ? ? ? ?? ? ? ? ??+?+ = ? ? ? ? + ? ? = ? ? (2.67) . )( ))(1( 12 1)( ;)(11)( 12 11 12 ?? ? ? ?= ? ? ??= ? tCPR tCPR tSMM tCPRtSMM (2.68) Also C(t) will be affected by the introduced uncertainty in WAC(t): {})()()](1)[()1()( tSMMtgtSMMtAtBtc +??= , (2.69) where ); 12 )( 1()( ; )12/)(1(1 12/)( )( tWAC tg tWAC tWAC tA tWAM += +? = +? (2.70) and {} ,)( )( )](1[ )( )1( )]()()[1( )( )()()](1)[( )1()( ? ? ? ? ? ? ? ? +? ? ? ?+ +?? ? ? ++? ? ?? = ? ? tSMM tg tSMM tA tB tgtAtB tSMM tSMMtgtSMMtA tBtc ?? ??? (2.71) where ; )( 12 1)( ; )( ] 12/)(1 )12/)(1())12/)(1(1( )( 144 1 )12/)(1(1 1 12 1 [ )( 2 ?? ? ? ? ? = ? ? ? + +? ++? + +? = ? ? +?+? +? tWACtg tWAC tWAC tWAM tWACtWAC tWAC tWAC tA tWAMtWAM tWAM (2.72) 40 And the PA gradient estimator for balance B(t) is as the following: . )( 12 )1()( ) 12 )( 1( )1()( );() 12 )( 1)(1()( ???? ? ?? + ? ? ?+ ? ?? = ? ? ?+?= tWACtBtctWACtBtB tc tWAC tBtB (2.73) The PA estimator for the discounting factor is unchanged, so we can get the modified Fourier duration and volatility duration. 41 2.4 Numerical Example 2.4.1 Specification of Numerical Example We need to specify two sets of data to price the mortgage: the mortgage data and the interest rate data, which includes the initial term structure and parameters for the interest rate model. We price different mortgages to examine the different impacts that a term structure shift or change in volatility may have on different mortgage products. The following data are fixed for all products: Unpaid Balance/UPB =$4,000,000; WAM =360 months. Table 2.1 shows the difference between all the products. All the ARM products have the same subsequent adjustment period of 12 months, period cap/floor of 0.02, lifetime cap of initial WAC plus 0.06, and no lifetime floor. Product WAC Index Adjust First FRM 0.07425 N/A N/A 1 Year ARM 0.06425 Treasury 1 Year 12 month 3/1 FP 1 ARM 0.07425 Treasury 1 Year 36 month 5/1 FPARM 0.07425 Treasury 1 Year 60 month 7/1 FPARM 0.07425 Treasury 1 Year 84 month 10/1 FPARM 0.07425 Treasury 1 Year 120 month 1 Year ARM 2 0.07425 Treasury 10 Year 12 month Table 2.1 Product Specification for Mortgage Pricing 1 FP ARM refers to Fixed Period ARM, which keep the coupon rate constant for a certain period, and then adjust periodically, generally once a year. So All the FP ARM products are the same, except different Adjust First date, which is the first coupon reset date. 2 This ARM is not a mortgage product in the market at present, and is constructed for illustration purpose only. The following sections will discuss why we introduce this product, and what nice properties it has. 42 We use the same parameters for all the different products in order to have comparable results. Thus we set all the products to have the same coupon rate, except the first 1 year ARM with index of Treasury 1 year rate, which has a 100 basis points (bps) teaser rate. All the ARM products have the same characteristics, except for the Adjust First date, which is the feature that distinguishes these products. Our initial term structure is the following: f(0,t)=ln(150+12t)/100, t=0,1,?,360. This will produce an upward-sloping curve increasing gradually from 5% to 8.7% along 30 year maturity, and R(0,t) is acquired by calculating the following: );0,0()0()0,0(, ),0( ),0( 0 frR t duuf tR t === ? (2.75) which increases from 5%, to 7.78% gradually. Our assumptions for interest rate model parameters are the following: a=0.1; ?=0.1; ? n =0.00025, n=0,1,2,3 (used in the FD gradient and gamma estimator calculation); ??=0.00025, (used in the FD vega estimator calculation). 2.4.2 Comparison of PA and FD gradient estimators In order to test whether our PA gradient estimators are accurate, and are within the error tolerance range, we calculate the finite difference (FD) gradient estimators at the same time during our pricing process. This section will demonstrate the accuracy of our PA estimators of delta, vega, and gamma for FRM, as well as the delta and gamma for ARM. 43 Comparison of Modified Fourier Gradient Estimators for FRM Figure 2.5 shows the FD estimator, PA estimator, their difference, and standard deviation of their difference for n td ?? ? )( . The four curves in each chart are specified as following, which will be the convention for the rest of the paper: Blue: Modified Fourier Order 1; Green: Modified Fourier Order 2; Red: Modified Fourier Order 3; Cyan: Modified Fourier Order 4. We can see that although these two estimators are pretty close, there exists a pattern in the difference of these two estimators. This will be explained later in the error analysis section. Figure 2.6 shows the PA and FD gradient estimators for cash flow c(t): they are pretty close, and the difference behaves as random noise. Based on ? ? ? ? ),(tc and ? ? ? ? ),(td , we can calculate ? ? d tdPV ),( , and figure 5.3 shows us the n d tdPV ? )( . Figure 5.4 shows the 95% confidence interval for difference between PA and FD estimators of n d tdPV ? )( , and we can see that 0 is generally contained in the 95% confidence interval. 44 Figure 2.5 Gradient Estimator Comparison for ?d(t)/ ?? n Figure 2.6 Gradient Estimator Comparison for ?c(t)/ ?? n 45 Figure 2.7 Gradient Estimator Comparison for dPV(t)/ d? n Figure 2.8 95% Confidence Interval for dPV(t)/d? n 46 Comparison of Vega Estimators for FRM In this section, we also compare the FD and PA estimators for the gradient w.r.t. interest rate volatility: Vega. Figure 2.9 shows the FD estimator, PA estimator, their difference, and standard deviation of their difference for ?? ? )(td . Also there exists a pattern in the difference of these two estimators. This will also be explained later in the error analysis section. Figure 2.10 shows the gradient estimators for cash flow c(t): they are pretty close, and the difference behaves as random noise. Figure 2.11 shows us the ?d tdPV )( , and figure 2.12 shows the 95% confidence interval for ?d tdPV )( , and we can see that 0 is always contained in the 95% confidence interval. Figure 2.9 Gradient Estimator Comparison for ?d(t)/?? 47 Figure 2.10 Gradient Estimator Comparison for ?c(t)/?? Figure 2.11 Gradient Estimator Comparison for dPV(t)/d? 48 Figure 2.12 95% Confidence Interval for dPV(t)/d? Comparison of Gamma Estimators for FRM For gamma estimation, ?=[? 1 ? 2 ? 3 ? 4 ]?. So 2 2 )( ?? ? td , 2 2 )( ?? ? tc , or 2 2 )( ?? ? tPV is a 4x4 matrix. If we want to estimate this matrix by the FD method, we would need 144 points to estimate 48 first order derivatives and to estimate 16 second order derivatives. The following figures show the FD, PA estimators, the difference and STD of difference for diagonal gamma elements. 49 Figure 2.13 gamma estimators for 2 2 )( i td ?? ? , i=1, 2, 3, 4 Figure 2.14 gamma estimators for 2 2 )( i tCPR ?? ? , i=1, 2, 3, 4 50 Figure 2.15 gamma estimators for 2 2 )( i tCF ?? ? , i=1, 2, 3, 4 Figure 2.16 gamma estimators for 2 2 )( i tPV ?? ? , i=1, 2, 3, 4 51 Comparison of ARM gradient estimators For ARM products, we basically have the same set of PA gradient estimators to compare with FD gradient estimators, with one additional set of estimators for i tWAC ?? ? )( (figure 2.17). To illustrate the accuracy of our simulation in a brief way, we only show the FD and PA gradient estimator comparison for one ARM product, 1-Year ARM with index of 1-Year Treasury rate, adjusted annually. Figure 2.17 Gradient Estimator Comparison for i tWAC ?? ? )( , i=1, 2, 3, 4 Figures 2.18 and 2.19 show the FD/PA gradient estimator comparison for i tPV ?? ? )( and ?? ? )(tPV for this ARM product, respectively. 52 Figure 2.18 Gradient Estimator Comparison for i tPV ?? ? )( , i=1, 2, 3, 4 Figure 2.19 Gradient Estimator Comparison for ?? ? )(tPV 53 2.4.3 Result Analysis Efficiency Analysis In financial practice, people are more interested in duration, which is the percentage change for a security, once there is a minor shift in one parameter, which mathematically is expressed as NPVd dNPV duration 1)( ? ? = . (2.76) Actually, there should be a minus sign before the expression, since the original duration of fixed income securities measures the percentage price drop resulting from an increase in the interest rate. Yet for our analytical purpose, we do not need the duration always to be positive, since from the following numbers, we see that durations can also be negative. Table 2.2 shows the FD and PA durations for FRM30, their 95% confidence interval, and the error range of the mean. Fourier Order 0 1 2 3 Vega PA estimator -6.4816?0.1017 3.1012?0.1860 -0.5705?0.1817 0.6269?0.1189 -6.7567?0.6712 FD estimator -6.4814?0.1017 3.1001?0.1860 -0.5695?0.1816 0.6259?0.1188 -6.7565?0.6712 Absolute Error -0.0002 0.0011 -0.001 0.001 -0.0002 Relative Error 0.0031% 0.0355% 0.1753% 0.1595% 0.0030% Table 2.2 Comparison of PA/FD Duration We can see that the error size is very small, and the 95% confidence intervals are almost the same. Thus from the accuracy point of view, we can use PA estimator to replace FD estimator without causing too much problem. And the improvement in computation efficiency is enormous. The FD duration estimator works in the following way: 54 )( 1 2 )()(1 * ),( ?? ???? ? ? NPV NPVNPV NPVd tdNPV ????+ = . (2.77) Thus for each parameter, we need two additional simulations. In our case, we need 2x5+1=11 simulations to estimate the FD duration. However, by PA estimator, we only need one simulation. Ignoring the costs of middle steps, and middle variables, we can reduce the computational time by 10/11, or 90.9%. When we consider the second order derivative, gamma, the computational efficiency improves even more. The following table shows the comparison of convexity estimators for FRM. Convexity is calculated as following: NPVd NPVd convexity 1)( 2 2 ? ? = . (2.78) As we have mentioned earlier, we only estimated part of the FD gamma estimators, via using the PA delta estimators. Because to fully estimate one set of 25 gamma estimators, we would need to simulate 225 times to get all of them. And each element is a 360 by 300 (time length by simulation path) matrix. So from the above analysis, we can see that by the conventional FD method, to estimate one full set of duration and convexity estimators with 5 free variables, would require 11 plus 225 simulations. Since we achieve almost the same accuracy by a single simulation in PA analysis, the simulation cost is reduced roughly by more than 99.5%. However, we also need to contemplate the introduced costs of intermediate variables as a tradeoff of the PA method. 55 Convexity = Gamma/Mortgage Value Mortgage Value = 4.22E+08 FD estimator 0 1 2 3 Vol 0 -246.5944896 N/A N/A N/A N/A 1 N/A -1871.407927 N/A N/A N/A 2 N/A N/A -1854.492905 N/A N/A 3 N/A N/A N/A -2000.544882 N/A Vol N/A N/A N/A N/A -4751.605032 PA estimator 0 -246.6418706 951.9319609 -646.2296558 161.8535453 919.0969179 1 951.9319609 -1871.360546 1251.356282 -233.9200682 -1435.431523 2 -646.2296558 1251.356282 -1854.208619 1106.51252 1223.425174 3 161.8535453 -233.9200682 1106.51252 -2000.92393 -715.5006989 Vol 919.1916799 -1435.407832 1223.377793 -715.4533179 -4755.158608 Fourier Order 0 1 2 3 Vol PA estimator -246.5944896 -1871.407927 -1854.492905 -2000.544882 -4751.605032 FD estimator -246.6418706 -1871.360546 -1854.208619 -2000.92393 -4755.158608 Absolute Error 0.047381014 -0.047381014 -0.284286087 0.379048115 3.553576082 Relative Error 0.0192% 0.0025% 0.0153% 0.0189% 0.0748% Table 2.3 Comparison of Convexity Estimators We did all the simulations on a Pentium III 800 MHz processor, with 512 MB memory, in Matlab Release 12.0 under Windows 2000. Here is the simulation comparison. Method FD PA Memory Required 17 MB 54 MB Simulation Time for 300 paths 115.5 765.8 Number of Duration Measures 5 5 Simulation required for estimating Duration 11 1 Number of Convexity Measures 25 25 Simulation required for estimating Convexity 225 1 Total Simulation 236 1 Total Expected Simulation Time 27257.7 765.8 Efficiency Improvement 97.2% Table 2.4 Comparison of Computing Costs 56 Accuracy Analysis In order to validate the predictive power of our PA estimator, we setup a test case to compare the predicted percentage change in the MBS price with the real percentage change. The test case is set up as following: .55 ;3,2,1,0,55 ;))1(cos(00 3 0 / 0 ?=? =?=? ??+= ? = ? e ne en ,t)R(,t)R(Perturbed_ n n Tt n ? ? (2.79) . 004--5.0474e = ? = ? NPV NPVNPVPerturbed_ NPV NPV While the predicted change in NPV is calculated as following: 004--5.0414e ' 2 1 ' )(' 2 1 )'( 2 2 = ????+??= ?? ? ? ?? + ?? ? ? ? ? ??? ? ? ?? ? convexityduration NPV NPV NPV NPV NPV NPV (2.80) where ??=[? 1 ? 2 ? 3 ? 4 ??]. We can see that the relative error by using both duration and convexity measures is only 0.0056%, while using duration measures only would produce a relative error of 0.1403%. So this test validates the predictive power of our PA gradient estimators. In the next section, we are going to show that PA estimator not only is more efficient than FD estimator, but also is a more accurate estimator. 57 Error Analysis Figure 2.5 and 2.9 show that there exists a pattern in the difference of gradient estimator of discounting factor d(t). Actually this has two reasons: the calculation of forward rates f(0,t) and the finite difference estimator of d(t). This could be verified by figure 2.9, which shows the difference of FD and PA n tf ?? ? ),0( estimators. We know that in the Hull-White model, f(0,t) is determined by (2.23). However, generally we do not have an explicit function form for R(0,t). Instead, we only have discrete points for term structure, so R(0,t) is estimated by interpolation. And f(0,t) is further estimated by calculating the difference between adjacent points on R(0,t) as t tR ? ? ),0( , which is not so accurate. The detailed calculation is given below. Figure 2.20 Difference of FD/PA ?f(0, t)/ ?? n estimators 58 ? ? ? ? ? ? ? ? ? = ? ??? << ? ????+ = ? ?? = ? ? termmaximum theis T ,, ),0(),0( 0, 2 ),0(),0( 0, )0,0(),0( ),0( Tt t tTRTR Tt t ttRttR t t RtR t tR (2.81) So using FD method to calculate the f(0,t) will result inaccuracy in FD estimator of n tr ?? ? )( , and this will result inaccuracy in d(t). Also we know that d(t) takes the following form: }])([exp{)( 1 0 tirtd t i ??= ? ? = , and t ir tdt ir tir td t i t i t i ? ? ? ?=? ? ? ???= ? ? ??? ? = ? = ? = ) )( ()() )( (}])([exp{ )( 1 0 1 0 1 0 ??? . (2.82) Figure 2.21 Fuction of xe -x 59 However, when we use FD method to estimate the first order derivative of e -x , the FD estimator is always greater in the absolute value, because e -x is a convex function. So FD estimator of ?? ? )(td is always biased, the bias decreases as the FD step width reduces. The bias increases linearly, while d(t) decreases exponentially. As a result, the bias takes the form of xe -x . Compare the difference of FD/PA ?? ? )(td gradient estimators and the figure of xe -x as in Figure 2.14, which resembles the error pattern very closely. For the PA method, n tf ?? ? ),0( is estimated by the following formula, 0 / // )( ))1(sin())1(cos( ),0( 0 00 T en enen tf Tt TtTt n ? ?? ?+?= ?? ? ? ?? (2.83) which does not involve the FD estimation of t tR ? ? ),0( . And ?? ? )(td is directly estimated using its analytical form of first order derivative. So the PA estimator is more accurate than the FD estimator. 60 2.5 Interpretation of the Results In this section, we briefly present the durations for various mortgage products, which show different trends for modified Fourier duration of different order. And we try to interpret how the modified Fourier shocks of different order would affect the discounting factors and the cash flows, and then the present value (PV) of the mortgage. Then we analyze the relationship of mortgage prepayment option and mortgage duration. Based on these analysis, we propose a potential new ARM product, which could reduce the duration over any of the existing mortgages, while having a less volatile index than most existing mortgages. This product would benefit both the investors who want to reduce the interest risk, and the mortgage borrowers who want to have a fairly stable coupon rate. 2.5.1 Overview of the Results The following table shows the durations for various ARM and FRM products we specified and priced in section 2.4: Fourier Order 0 1 2 3 Vega ARM TSY 1 -1.7761 -4.313 6.5674 -2.3822 -3.4618 FP 3/1 ARM -2.8441 -2.9642 7.1814 -3.4784 -4.1601 FP 5/1 ARM -3.8514 -1.1609 5.9506 -5.3355 -5.2456 FP 7/1 ARM -4.3054 -0.3064 4.9272 -5.4472 -5.6651 FP 10/1 ARM -5.4256 1.6401 1.7592 -2.6933 -6.6163 FRM30 -6.4816 3.1012 -0.5705 0.6269 -6.7567 Table 2.5 Durations of Different Products The relation can be better illustrated with the figure 6.1. The zero th order modified Fourier duration (with respect to ? 0 ) is the same as Option Adjusted Duration (OAD), which measures the price percentage change to a parallel interest term structure shift. Other modified Fourier durations are the same measure, with respect to other interest 61 term structure changes. Vega measures the price percentage change to an interest volatility change. As we can see, for OAD and Vega, the most important hedge measures, FRM30 has the highest numbers, and 1-Year ARM has the lowest. For everything between pure FRM and pure ARM, there exists a monotonic relationship with the product?s approximation to an FRM30. For example, the Fixed Period 10/1 ARM is more like an FRM30 than a Fixed Period 7/1 ARM, so it has higher OAD, and higher Vega. This means that ARM products have a lower interest risk than FRM products, since an ARM borrower takes more interest risk than an FRM borrower. This result is consistent with Kau et al.[1990,1992,1993] and Chiang [1997]. However, an interesting phenomenon is that the first order modified Fourier duration (with respect to ? 1 ) actually decreases, and changes sign as volatility of the coupon rate decreases. This indicates that an opposite move of the long-term and short-term rates would not only affect ARMs with a different magnitude, but also has a reverse effect from FRMs. Here is the explanation for this. The first order modified Fourier duration models the following changes in term structure: ? Short-term rate increases; ? Intermediate term rate (e.g. 10 year rate) doesn?t change, or moves only a little bit; ? Long-term rate decreases. In this scenario, people with a short-term ARM, e.g. 1-year ARM are burnt the hardest, so they are going to refinance anyway, even if the prevailing mortgage rate does not change a lot. This will create huge prepayment, and reduce the NPV of the ARM mortgage. People with FRM, on the other hand, have no incentive to refinance, since the refinance mortgage rate (highly correlated with 10-Year Treasury rate) does not change a 62 lot. This will make the future cash flow more stable and valuable, since they are discounted at a lower long-term interest rate, and increase the NPV of the FRM mortgage. Figure 2.22 Duration vs. Products The above analysis is based mainly on intuition, and does not show how will this term structure shock affect the discounting factors, cash flows, and NPV of MBS. In the following section, we will see what effect each one of the modified Fourier functions has on these components of MBS for various mortgage products. 2.5.2 Modified Fourier Shock Impact The following 8 charts will show different modified Fourier shocks on term structure R(0,t), and their impact on d(t), CF(t), and PV(t). Duration vs. Products -8 -6 -4 -2 0 2 4 6 8 AR M T S Y 1 FP 3/ 1 ARM F P 5 / 1 A RM F P 7 / 1 A RM F P 10 / 1 A RM FRM3 0 Products Duration Harm Duration 0 Harm Duration 1 Harm Duration 2 Harm Duration 3 Vega 63 Figure 2.23 The Impact of Modified Fourier Function Order 0 on FRM30 Explanation: A parallel shift in the upward slope term structure will have a negative impact on the discounting factor. Also people are less likely to prepay in the near future, which reduces the cash flow in the short term, and increase the cash flow in the long term a little bit. However, the overall effect of such a shift on present value is negative, and thus reduces the NPV of this MBS. 64 Figure 2.24 The Impact of Modified Fourier Order 1 on FRM30 Explanation: A shift of this shape in the upward slope term structure will have a mixed impact on the discounting factor: decrease it in the short term, but increase it in the long term. Also people are more likely to prepay in the near future, which increases the cash flow in the short term, and reduces the cash flow in the long term a little bit. However, the overall effect of such a shift on present value is positive, and thus increases the NPV of this MBS. 65 Figure 2.25 The Impact of Modified Fourier Order 2 on FRM30 Explanation: A shift of this shape in the upward slope term structure will have a mixed impact on the discounting factor: increase it in the middle term, but decrease it in the long term. There is little incentive for people to prepay in the near future, and they will also cling to their current coupon rate in the middle term, because at that time the refinance rate will increase. However, the overall effect of such a shift on present value is cancelled out, and has little impact on the NPV of this MBS. 66 Figure 2.26 The Impact of Modified Fourier Order 3 on FRM30 Explanation: same as Modified Fourier Order 2 67 Figure 2.27 The Impact of Modified Fourier Order 0 on ARM TSY 1 Explanation: A parallel shift in the upward slope term structure will have a negative impact on the discounting factor. Also people with ARM are less likely to prepay in the near future, because they have a lower ARM rate than the refinance. Yet they will start prepay in the middle term, because short term rate at that time will increase, due to the upward slop term structure. This behavior will reduce the cash flow in the short term, and increase the cash flow in the middle term. However, the overall effect of such a shift on present value is negative, and thus reduces the NPV of this MBS. Yet the impact will be much smaller than that for FRM. 68 Figure 2.28 The Impact of Modified Fourier Order 1 on ARM TSY 1 Explanation: A shift of this shape in the upward slope term structure will have a mixed impact on the discounting factor: decrease it in the short term, but increase it in the long term. Also people are more likely to prepay in the near future, which increase the cash flow in the short term, and reduce the cash flow in the long term a little bit. The overall effect of such a shift on present value is negative, and thus decreases the NPV of this MBS. 69 Figure 2.29 The Impact of Modified Fourier Order 2 on ARM TSY 1 Explanation: A shift of this shape in the upward slope term structure will have a mixed impact on the discounting factor: increase it in the middle term, but decrease it in the long term. People will cling to their low ARM rate for the first few years, but then start to prepay in the middle term, since short term rate will increase at that time. The overall effect of such a shift on present value is positive, due to the increase cash flow and discounting factor in the middle term. 70 Figure 2.30 The Impact of Modified Fourier Order 3 on ARM TSY 1 Explanation: much like Modified Fourier Order 3, yet because the reverse effect of discounting factor, the overall effect will be negative. 6.3 Potential New ARM Product Duration is used to measure the interest risk of a fixed income security. The higher the duration is, the more interest risk that security bears. From the investor?s perspective, she will benefit if interest rates fall, and suffer if interest rates climb, if the security is non-callable (no prepayment option). From the mortgage borrower?s point of view, he will exercise his prepayment option if interest rates drop, and thus reduce the benefit for the investor. He will be able to lock in the low mortgage rate (for FRM), in case interest rates climb, and thus hurt the investor more. However, for the ARM borrower, he benefits from the rate drop, so he does not prepay like the FRM; thus the 71 MBS investor will also benefit. And he also pays the high coupon rate when interest rates increase, and the ARM MBS investor will not suffer like the FRM MBS investors. From this perspective, the ARM should have a lower duration compared to FRM. ARM borrower?s coupon rate fluctuates with the current interest rate, which is correlated with the prevailing mortgage rate. Because of this, she will have less incentive to prepay when interest rate drops. So the prepayment option value for a FRM borrower will be larger than that of an ARM borrower. This is compatible with the market, where FRM mortgages are sold with the highest rate (borrower pays for the valuable prepayment option), and ARM, that adjust most frequently are offered with the lowest rate. In option theory, we know that option value generally increases as the volatility of underlying asset increases. However, from the above analysis, we also know that the option value for a FRM is generally greater than for an ARM, while an ARM bears a more volatile coupon rate than a FRM. This looks like a contradiction to the option- volatility relationship. In fact, it?s not, because the underlying asset of a prepayment option is not its coupon rate, but the difference between the coupon rate and the prevailing mortgage rate. In most cases, the more volatile the coupon rate is, the less the difference will be, and the less valuable the option will be. However, the borrower does not like the volatility, which put her at risk when interest rate jumps. The investor, on the other hand, does not like the prepayment, which reduce her investment value. It seems that no product can both reduce the coupon rate volatility and the prepayment option at the same time. Is this true? We will see that we can achieve both goals in a potential new ARM product. 72 We have mentioned that the underlying asset for the prepayment function is the spread between the coupon rate and the prevailing mortgage rate. So an ARM bearing a volatile index does not necessarily indicate a less volatile spread. From historical data, we know that 10-Year Treasury rate is highly correlated with conventional (FRM30) mortgage rate. Figure 6.10 shows the two rates for the period between 1971 and 2001. The correlation calculated is 97.9%. Figure 6.10 also shows the 10-Year Treasury Rate and the 1-Year Treasury rate, which is the most commonly used index in ARM. As we can see, the 1-Year Treasury rate is relatively more volatile than the 10-Year Treasury Rate. The calculated standard deviation is 2.7890 for the 1-Year Treasury rate, and 2.6309 for the 10-Year Treasury rate. However, the standard deviation of spread of FRM30 vs. TSY10 is 0.58, compared with the standard deviation of spread of FRM30 vs. TSY1 at 1.16. Obviously 10-Year Treasury rate has a lower volatility and also a lower volatile spread. The spread between conventional mortgage rate (FRM30) and 10-Year Treasury rate and the spread between conventional mortgage rate (FRM30) and 1-Year Treasury rate are also shown, which indicates that ARM with index of 1-Year Treasury rate has a more volatile spread. 73 Figure 2.31 10-Year T Rate, 1-Year T Rate, and mortgage rate Thus if we construct an ARM with index of 10-Year Treasury rate, and reset it more frequently, we could expect a lower duration. So we construct such an ARM with the adjustment period of 12 months. This ARM does not exist at present; it is for illustration purposes only. We then got the modified Fourier duration measures as following: Fourier Order 0 1 2 3 Vega ARM TSY 10 -1.2741 -4.0635 3.1819 -0.4894 -1.8855 Figure 2.32 shows the new ARM product?s duration against duration of other mortgage products we calculated earlier. We compare this set of durations with table 6.1, 74 and we can see that this product has the smallest durations for modified Fourier function order 0 and 3, as well as for vega. The durations for modified Fourier function order 1 and 2 are not very high. And we know that generally when there is a shock on the term structure, the biggest magnitude would be that of the first-order modified Fourier function, and volatility is also a big impact. So this product would actually have the least percentage change during a common term structure shift, which satisfies the needs of investors. Duration vs. Products -10 -8 -6 -4 -2 0 2 4 6 8 A RM TS Y 10 ARM T S Y 1 FP 3/1 ARMFP 5/1 ARMFP 7/1 A RM F P 1 0/ 1 A RM F RM 30 Products Duration Harm Duration 0 Harm Duration 1 Harm Duration 2 Harm Duration 3 Vega Figure 2.32 New ARM TSY 10 Durations So we could predict that if there exist such a mortgage, it would have the least refinancing incentive, which would be a better product to suit investors? needs, and it will also have a less volatile index, which suits borrowers? needs. 75 2.6 Conclusion This paper applies perturbation analysis (PA) method to estimate MBS sensitivities. The sensitivity estimators include most interest risk measures like duration (equivalent to delta), convexity (equivalent to gamma), and vega. MBS products covered includes fixed rate mortgages (FRMs) and adjustable rate mortgages (ARMs). We first derive a general framework to derive the PA estimators of MBS, without restriction to MBS type, interest rate model, or prepayment model. Then we apply the PA estimator to both FRM and ARM products, in the setup of a one-factor Hull-White model and a commonly used prepayment model. We compare the PA estimators with finite difference (FD) estimators, and find that PA method can achieve at least the same accuracy as FD method, with a much lower computational cost. In the case we presented, the computational time is reduced by 95.7%, while the memory requirement increases only by a factor of 3, which can be handled by current computer technology with ease. Then we analyze the results of PA estimated sensitivity measures for various MBS products. We justify why and how different term structure shock would affect FRM and ARM differently. Based these analysis, we propose a potential new ARM product which could benefit both the MBS investor and the mortgage borrower. Future research includes applying this method to other MBS-like securities, since the PA method proposed in section 3 is a very general framework. These include other asset-backed securities, e.g. securities backed by student loans, car loans, credit card receivables. It is pretty straightforward to expand this framework to those securities, since all that is required is to apply a specific interest rate model and prepayment model. 76 Another area for further research is to incorporate more complicated prepayment and/or default models into the MBS pricing scheme. For MBS investors, the major concerns are price sensitivities to interest changes, which we have covered in detail. However, the MBS guarantor/insurer and issuer might have other concerns, e.g., how will the interest rate change affect the default behavior of the mortgage borrowers? Our framework would be able to serve this purpose as well. By applying the default model that same way as we apply a prepayment model, the default cash flow will take the place of payment cash flow, so the default cost sensitivities could be easily estimated. 77 Chapter 3 Hedging MBS in HJM Framework 3.1 Motivation As we have pointed out in our first essay, short term rate and long term rate do not always move in the same direction, it is sometimes misleading to use the conventional interest rate risk measures like duration and convexity to hedge fixed income instruments, especially MBS. One recent event can illustrate this point very well. In July 2003, Federal Reserve lowered the short term interest rate by another 25 bps, yet just in one month, the long term 10 year rate jumped upward for more than 100 bps. Part of the reason is that the rate deduction is lower than market expectation, and market responded with a selling wave in the bond section. So using a duration measure, which assumes the yield curve moves in parallel, will produce significant hedging error. It is natural to hedge against the factors of which any yield curve shift can be decomposed. We use a series of exponentially decaying modified Fourier series to approximate any interest rate change in our first essay. However, this is purely for the generality of modeling convenience, and there is no empirical evidence that such a series provides a good match of the actual yield curve shift. 78 A lot of literature studying the dynamics of interest rates found that there are three major factors affecting the yield curve: level, slope, and curvature. A common method to estimate these factors is Principal Components Analysis (PCA). See details in Litterman and Scheikman [1991], Litterman, Scheikman, and Weiss [1991), Knez, Litterman, and Scheikman [1994], Nunes and Webber [1997]. Despite the abundance of research on identifying the various factors affecting bond prices, there has been little research on hedging these factors effectively. Golub and Tilman [1999] compared different risk measures, like PCA, VaR, and key rate duration for yield curve risk, but did not give hedging performance for these different measures. In mortgage industry, practitioners generally use effective duration, and empirical duration in hedging. Goodman and Ho [1999] examined the performance of three different hedge ratios: effective duration, empirical duration, and option-implied duration, which is acquired from forward option for a given pass-through MBS. They found that the average hedging error for a monthly hedge could reach 120 bps in an 18-month period. And for a daily hedging, it is 25 bps in the same time period. They concluded that option-implied duration performs the best. However, it does not always outperform the other hedging measure, and the difference is small. Hayre and Chang [1999] compared effective duration and empirical duration, and found that effective duration calculated from OAS model are generally longer than empirical duration, and they challenged a few assumptions for effective duration calculation in OAS mode. To cite a few, the parallel yield curve shifts, absence of convexity, etc. These are also issues we addressed in our 79 paper, but in a more systematic way. They proposed a combined duration, which is a effective duration adjusted for correlations between changes in the yield and prices of MBS in recent market data, i.e., a combination of effective duration and empirical duration. There has not been a unifying framework in hedging MBS with factors affecting the yield curve shifts, and we would like to pursue in this direction, since we are pretty confident about its effectiveness in reducing the hedging error, and/or reducing hedging frequency. In order to incorporate these factors into MBS hedging strategy, we need to choose an interest rate model, which can handle these factors readily. HJM model is a good choice, because it is basically driven by volatility structure, and the volatility factors can take any shape, which easily accommodate the PCA factors we identified from historical data. In the rest of this essay, we discuss how to get volatility factors from historical interest rate data via the PCA method. In section 3, we are give the detailed implementation of HJM model with these estimated volatility factors. Then we derive the PA estimators for hedging MBS, which is very similar to Chapter 2, and we will not go into details to derive PA estimator for each state variable. In section 5, we give the detailed hedging algorithm with these hedging measures, and we discuss the performance of our hedging method in section 6. Section 7 concludes the essay, and gives potential future research directions. 80 3.2 Estimation of Volatility Factors via PCA The Principal Components Analysis method is generally used to find the explanatory factors that maximize successive contributions to the variance, effectively explaining variations as a diagonal matrix. This method has been used in yield curve analysis for more than 10 years, see Litterman and Scheinkman [1991], Steeley [1990], Carverhill and Strickland [1992]. Here we give a brief description of PCA method applied in yield curve analysis: 1. Suppose we have observation of interest rates )( jt i r ? at time t i , i=1, 2, ?, n+1, for different maturity dates ? j . 2. Calculate the difference )()( 1 , jtjtji ii rrd ?? ?= + , where the d i,j are regarded as observations of a random variable, d j , that measures the successive variations in the term structure. 3. Find the covariance matrix ),...,cov( 1 k dd=? . Write ),cov( where},{ ,, jijiji dd=??=? . 4. Find an orthogonal matrix P such that P?=P -1 and kk ? ... ?), ..., ?diag(?PP ??=? 11 where,'. 5. The column vectors of P are the principal components. 6. Using P, each observation of d j can be decomposed into a linear combination of the principal components. By setting jii dpe '= , where p i is the i th column of P, we can find e i , which is the corresponding coefficient for principal component i, i=1, ?, k. A 81 small change in e i will cause the term structure to alter by a multiple of p i along the time horizon. We use the weekly data of nominal zero coupon yield from January 1997 to October 2001 as the term structure data. All data were retrieved from Professor McLulloch?s web site at the Department of Economics, Ohio State University, at . For each observation date, interest rates are provided for maturities in monthly increments from the instantaneous rate to the 40-year rate, providing a total of 481 interest rates as principal components. Table 3.1 lists the eigen-values and % variance explained by the first ten factors, and Figure 3.1 graphs the shapes of the first four factors. Factor Eigenvalue Explained(%) Cumulative(%) 1 16.38 75.824 75.824 2 4.41 20.432 96.257 3 0.72 3.335 99.592 4 0.087 0.40 99.995 5 0.00088 0.0041 99.999 6 8.67E-05 0.00040 99.9996 7 1.59E-05 7.4E-05 99.99966 8 4.20E-06 1.9E-05 99.99968 9 4.03E-06 1.9E-05 99.99970 10 3.67E-06 1.7E-05 99.99972 Table 3.1. Statistics for Principal Components 82 Figure 3.1 The first four principal components The statistics indicate that the first three factors explain about 99.6% of the yield curve changes, and the first four factors explain about 99.995% of the total variance of yield curve. These results are similar to findings by Litterman and Scheikman [1991], and Nunes and Webber [1997]. Figures 3.2 and 3.3 plot the matching results with three and four factors, respectively, for a monthly yield curve shift, as well as for an annual shift. The figures indicate that four factors provide a substantially improved match, both for the short term and the long term, over three factors, so in our model we will use four factors. Thus, hedging against these factors will lead to a considerably more stable portfolio, thereby reducing hedging transactions and its associated costs. 83 Figure 3.2 Match monthly yield curve shift Figure 3.3 Match annual yield curve shift 84 3.3 Simulation in HJM Framework This section gives the detailed implementation of HJM model, using the volatility factors identified in PCA analysis. We know that, in a multifactor HJM framework, the dynamics of instantaneous forward rate looks like: ? = ?+?= N k ktkt tdZTtdtTtmTtdf 1 )(),,(),,(),( ? , (3.1) where under no arbitrage assumption, the drift term is determined by volatility structure. ? ? = ??=? N k T t tktkt dtTtdtTtm 1 ),,(),,(),,( ???? . (3.2) Assume our volatility functions take the following form: ),(),,( TtPCTt kktk ?? =? , (3.) where PC i (t, T) is the principal components we get in last section; ? i is a parameter to be calibrated to market price of interest rate derivatives. Detailed Implementation: 1. Input data include f(0,T), the instantaneous forward curve, and ? k (t, T), which has a specified functional form fitting into our PCA factors. 2. Start loop for maturity, if we need 10 year rate for 30 years, we need maturity at 40 year; 3. Start of time step loop; 4. Start of ? loop, to calculate ? k (t, ?) from t to T; 85 5. Calculate ? k * (t, T)= ? T t k dt ??? ),( , using numerical integration technique; 6. Calculate m(t, T)= ? k (t, T)* ? k * (t, T); 7. Advance f(t, T) one more step, in our simulation, one month increment: ;),(),(),( ,? +?=?+ k ktk zTttTtmTttf ? where z t,k is a series of independent standard normal random variables. 8. End of time step loop; 9. Short rate r(t)=f(t,t); Long rate r 10 (t)= 10 ),( 10 ? +t t dtf ?? ; 10. End of maturity step loop. 86 3.4 Deriving PA estimators in HJM Framework Following the logic in Chapter 2, we only need to derive the PA estimator for short rate r(t) and 10-year rate r 10 (t), since our prepayment model and valuation model are totally dependent on these two factors. If we assume that in a short period of time, the principal components for yield curve volatility are going to be constant, then any interest rate yield curve shift can be decomposed of these principal components, which is to say: ? ?=? k kk tPCtR )(),0( , (3.4) which is analogous to (2.35). Following the same logics as in (2.51), we can have the following: ).( )(),0( , ),0(),0( ),0(),0(),0( ),0( ),0( ),0( 2 2 tPC t tPC t tf tR t tR t tR t tR t t tRt tRt tR t tf k k k + ? ? = ?? ? ? ? + ?? ? = ? ? + ?? ? + ? ? ? ? = ? ? + ? ? ? ? ? ? ? ? ? ? = ? ? ?? ??? ??? (3.5) We know that in HJM framework: ),()( ttftr = , (3.6) So kk ttftr ?? ? = ?? ? ),()( . (3.7) 87 We also know: ? = ?+?= N i itit tdZTtdtTtmTtdf 1 )(),,(),,(),( ? . (3.8) When T=t+dt dt)df(t,ddt)f(t,tdt)dt,tf(t +++=++ . (3.9) So kkk dt)df(t,ddt)f(t,tdt)dt,tf(t ?? +? + ?? +? = ?? ++? ][ , (3.10) ? = ?? +? + ?? +? = ?? +? N i i k i kk tdZ dtttdtttmdt)df(t,d 1 )( ),(),(][ ? . (3.11) If we rewrite the drift term as: ),(*),(),( * 1 TtTtTtm k N k k ?? ? = = , (3.12) where ???? dtTt T t kk ? = ),(),( * . (3.13) Then gradient of m(t, T) can be written as ? = ? ? ? ? ? ? ? ? ? ? ?? ? + ?? ? = ?? ? N k k i ii k i k Tt TtTt TtTtm 1 * * ),( *),(),(* ),(),( ? ?? ? . (3.14) From the form of ),( Tt k ? ),(),,( TtPCTt kktk ?? =? . (3.15) 88 There is no direct relation ship between ),( Tt i ? and k ? 1 , so the above gradients are zero. This gives us kkk dt),df(dt)f(t,tdt)dt,tf(t ?? +? == ?? +? = ?? ++? ]0[ ... . (3.16) For the same reason, we can derive the 10 year rate gradient as: 10 ),( )( 10 10 ? + ?? ? = ?? ? t t k k d tf tr ? ? . (3.17) And follow the same logic, we can get the gradients of discounting factors, prepayment rate, cash flows, present values, etc. 1 Although observed k ? and k ? might have a positive correlation, i.e., when the volatility is high, the observed shift also might have bigger magnitude. But they have total different meaning, k ? is the parameter to calibrate to market price, and k ? is the observed shift in yield curve. 89 3.5 Hedging MBS in HJM Framework This section gives a detailed implementation of our hedging algorithm. Security to be hedged: MBS Hedging Instruments: Portfolio of {MBS, Treasury bonds with different maturities} Hedging Method: Dynamic hedging using PCA duration. vs. Conventional duration and convexity hedging Hedging Parameters: PCA duration Hedging Error: The net present value of the portfolio, which has initial value of zero Hedging Efficiency: Reduce hedging Error Hedging Strategy: Construct a portfolio, consisting of MBS and various T-notes, bonds, with 0 face value. Duration matched to 0. Rebalance at each time period to match the hedging parameters; compare the results with duration and convexity hedging. There are two issues we need to pay special attention to, in order to effectively execute the hedging strategy. Issue 1: With the coupon payment and prepayment of MBS, what needs to be done with this extra cash flow? Answer: Use this cash flow to rebalance the portfolio, basically to change the weights of Treasury bonds holdings. If the position is short in MBS, and long in Treasury bonds, we need to sell the Treasury bonds to honor the MBS payment. Issue 2: Some Treasury bonds used to hedging the MBS will expire before the MBS maturity date. This will hurt the capacity of available hedging instruments. 90 Answer: We only hedge the MBS for a short period of time, e.g. 3 years, and then we can use Treasury bonds with greater or equal to 3 years maturities. Another solution is to introduce on extra hedging instrument when there is one expiring at that period. Hedging Framework 1. At time 0, get the MBS price, gradients (PCA duration) by simulation (360x300 simulation needed). Zero coupon Treasury bonds price and gradients should be directly available from the yield curve, and the PCA factors; 2. Construct the portfolio, by shorting MBS to finance Treasury bonds; match the duration, and get the corresponding weights; 3. At time 1, use HJM model to update the yield curve, then get the new price and gradients of MBS as well as those of Treasury bonds; 4. Use MBS payment to rebalance portfolio (MBS payment is deterministic upon the last period yield curve); 5. Repeat 3, 4 for next month, till the end of hedging period; 6. Check the effectiveness of hedging strategy. Implementation of Hedging MBS with Treasury Bonds 1. Get mortgage information; 2. Get historical yield curve data; 3. Get Principal Components Factors; 4. Start clock for hedging period: m=0 5. Calculate MBS_Price(m), MBS_Duration(m) 4x1 , Payment(m), PrincipalPayment(m); 91 6. Choose hedging instrument: Treas_Portfolio=[12 36 60 84 120], each element represent months to maturity; 7. Calculate Treasury bond price Treas_Price(m) 5x1 , 5 hedging components are needed because of 5 factors to hedge: Price, and Duration 4x1 . Treas_Duration(m) 4x5 . 8. Solving for hedging ratio W(m): 15 15 55 )(_ )(_ )( )(_ )'(_ x x x mDurationMBS mpriceMBS mW mDurationTreas mpriceTreas ? ? ? ? ? ? = ? ? ? ? ? ? , if m=0; (3.18) 15 15 55 )(_ )1(_)1()'(_ )( )(_ )'(_ x x x mDurationMBS mpaymentMBSmWmpriceTreas mW mDurationTreas mpriceTreas ? ? ? ? ? ? ??? = ? ? ? ? ? ? , ifm>0. (3.19) 9. Calculate hedging error: )1(_)1()'(_)(_)( ?+??= mpaymentMBSmWmpriceTreasmpriceMBSmerror 10. Update loan.UPB=loan.UPB-PrincipalPayment(m); 11. Update loan.WAM=loan.WAM-1/12; 12. Update Treas_Portfolio=Treas_Portfolio-1; 13. m=m+1, go back to 5 until hedging period ends. 92 3.6 Hedging Performance Analysis In this section, we compare the hedging performance of our PCA-based hedging and traditional duration and convexity based hedging for a FRM30 MBS instrument. The principal balance of the MBS is $4 million. We are selling short this MBS at the market price, and use the proceeds to buy treasury bonds. Initial net present value of the hedging portfolio is zero. Every month, we try to rebalance the portfolio, and we sell part of our bonds to meet the payment obligation of the MBS. Hedging error is defined as the net present value of current portfolio at each time point. We carry on this practice for 22 months, during which our PCA estimation does not change dramatically. We repeat the hedging practice for 25 simulations, which is relatively few, because the simulation scheme takes an extremely long time. The PCA- based hedging takes around 40 CPU hours to finish, while the duration and convexity based hedging takes 120 CPU hours to complete the task. Figure 3.4 shows the hedging performance of three PCA factors, while Figure 3.5 shows the hedging performance of duration and convexity hedging. We can see that the standard deviation of PCA-based hedging ranges from $4000 to $20000, which is 10 bps to 50 bps for a $4 million portfolio. Consider the standard deviation of duration and convexity based hedging, which ranges from $60000 to $200000, i.e. 150 bps to 500 bps of the hedging balance. The hedging improvement is obvious. 93 Mean Hedging Error -5.00E+04 -4.00E+04 -3.00E+04 -2.00E+04 -1.00E+04 0.00E+00 1.00E+04 2.00E+04 3.00E+04 0 5 10 15 20 25 month Hedging Error mean_PCA mean_D&C Figure 3.4 Mean Hedging Error of PCA vs. D&C Standard Deviation of Hedging Error 0 50000 100000 150000 200000 250000 300000 0 5 10 15 20 25 month Hedging Error std_PCA std_D&C Figure 3.5 STD of Hedging Error: PCA vs. D&C 94 3.7 Conclusion In this essay, we proposed a new method to hedge the interest risk of MBS, based on PCA factors estimated from historical interest rate data. We estimated the PA estimators for hedging MBS, and implemented the hedging with a dynamically re- balancing portfolio of MBS and Treasury bonds. We achieved much better hedging efficiency, compared with traditional hedging, not only in the measure of mean hedging error, but also in the standard deviation of hedging error. We made the following contribution: ? A unified hedging framework for hedging yield curve shift and volatility factors; ? Improved hedging efficiency compared with traditional duration and convexity based hedging. Our monthly hedging get very close results to daily hedging with traditional hedging method. We would like to pursue in the following directions for our future research: ? Apply this hedging method to more sophisticated prepayment models, and analyze the robustness of this hedging algorithm; ? Improve computational efficiency of the algorithm, which is now very time consuming. 95 Chapter 4 Hedging the Credit Risk of MBS: A Random Coefficient Approach 4.1 Motivation In our previous two chapters, we have assumed that the credit risk of the MBS is totally absorbed by the MBS issuer, and the MBS investor only needs to hedge the interest rate risk due to voluntary prepayment, including housing turnover and refinancing. This assumption is reasonable since in the secondary market for conforming mortgages, the three major MBS issuers, Ginnie Mae, Fannie Mae, Freddie Mac 1 , all promise that they will guarantee the principal payment when there is a default event incurred on the mortgage borrower?s side. The MBS issuers have the following methods to mitigate the credit risk: ? Mortgage Collateral: Basically when a default occurs, the collateral property will become REO(Real Estate Owned), and the issuer can foreclose the mortgage and sell the property, and recover whatever is left; ? Primary Mortgage Insurance (PMI): If a borrower initiate a loan with LTV greater than 80%, she will be required to purchase mortgage insurance. If default occurs, the mortgage insurance company pays the owner of the mortgage whatever is promised in the insurance contract, generally 35% for a 95 LTV loan, and 20% for a 85 LTV loan; 1 However, the credit risks of these agencies are different. Ginnie Mae is guaranteed by the full faith and credit of the United States government. Both Fannie Mae and Freddie Mac have $2.2 billion line of credit with the Treasury department. Also they receive an implicit guarantee from the government, since most 96 ? Credit Enhancement: The MBS issuer can purchase additional insurance from a mortgage insurance company for a mortgage pool. This deal is also called pool insurance, or backend credit enhancement. It is not necessarily purchased from the same company that provides PMI in the mortgage pool. There is generally an auction among several insurance companies, and the bidder with the most competitive price will be awarded the contract. When hedging the credit risk of the MBS with credit enhancement from a third party, the issuer is now exposed to the credit risk of the counter party. In order to hedge the credit risks effectively and efficiently, we not only need to model the default behavior of the mortgage borrower, but also need to understand the credit worthiness of the counter party. The credit worthiness of a given counter party for a given time horizon is generally called a haircut 2 . We need to model the haircuts of the counter party to perform the following tasks: ? Calculate the insurance premium, i.e., the purchase price for the insurance policy, to be paid. Apparently, a company with lower credit risk should be charging higher fees, and vice versa, since lower credit risk means better insurance policy. ? Estimate the credit loss, and report it to external investors and regulators. Currently the Office of Federal Housing Enterprise Oversight (OFHEO), regulator of Fannie Mae and Freddie Mac, requires both GSEs to report their risk-based market participants believe that federal government will interfere whenever any of these two giant GSEs steps in financial distress. 2 This term is used to determine the reduction applied to promised payment, due to credit risk, e.g., a 25% haircut means that the promised payment needs to be reduced to 75%. 97 capital calculated by pre-specified haircuts for different rated counter parties. With the implementation of Basel Accord II 3 , internal credit risk models could be used to calculate the haircuts, and in-house model for calculating the counter party credit risk is of extreme importance in reporting the credit risks. We below show that a haircut is actually a credit risk measure similar to credit spread. And estimation of a haircut is equivalent to estimation of credit spread. Suppose we need to take the haircut H(t) for a promised future payment of $1, what would be the price for this promised payment? In risk neutral probability, the price should be: P=exp(-r(0,t)*t)[1-H(t)] Where r(0,t) is the spot rate for maturity t. If the promised payment can be viewed as a zero-coupon defaultable bond with face value of $1, its price is given by P=exp{-[r(0,t)+CS(t)]*t}, where CS(t) is the credit spread for maturity t. Apparently haircut and credit spread have the following one-to-one relationship: H(t)=1-exp(-CS(t)*t) Once we estimated the credit spread of the third party?s defaultable bonds, we will get the haircut we need to impose on the insurance contract automatically. So it is of critical importance that we have a good estimation for the credit spread changes of the counter party. There are generally two ways to model the dynamics of the credit spread: 3 Basel Accord II is the new international banking regulation rule proposed by Basel Committee, which will be implemented before 2006. It gives more flexibility in treating credit risk, and internal credit risk models can be used in calculating risk-based capital requirement, which is the capital a financial institution needs to reserve in order to alleviate the credit risk exposure to counter parties. 98 theoretical approach and empirical approach. There has been a lot of published work on the literature on the theoretical part of credit risk modeling: either using structural models (Merton [1974], Longstaff and Schwartz [1995], Collin-Dufresne and Goldstein [2001]), or reduced form (hazard rate) models (Duffie and Singlton [1999], Madan and Unal [2000]). In empirical work, different models are estimated and fitted with market data, and the performances of these models are compared in recent papers, e.g. Eom, Helwege, and Huang [2003]. Recently there has been interest in using regression to determine the factors affecting credit spread changes, because neither structural nor reduced form models can handle the large number of factors affecting credit spread changes. With a flourishing credit derivative market, there is a great need for identifying the factors that affect credit spread, in order to find possible financial instruments to hedge credit derivatives written on credit spreads. The main model used in these researches is the simple linear regression model, e.g., Duffee [1998], Collin-Dufresne, Goldstein and Martin [2001], Huang and Kong [2003]. However, these models generally do not offer very compelling results. In this essay, we identify the theoretical drawbacks of this type of models, and address these problems with a new approach: the Random Coefficient Regression (RCR) model, which we can handle the non-constancy phenomena of credit spread sensitivities. The rest of this essay is organized as follows. We first give a literature review in the following section; specifically we are going to discuss several important papers. In section 3, we introduce the random coefficient regression (RCR) model is given and then 99 apply the model to estimate the dynamics of credit spread changes, using variables from the simplest structural model. Description of the data is given in section 5, and the regression results are discussed in the next section. We show that our assumption about non-linearity and non-constancy of credit spread changes are well supported by the regression results, also the regression results are consistent with theoretical structural models, such as Merton [1974]. In the last part of this essay, we give conclusions and possible future research directions. 100 4.2 Literature Review There has been a lot of recent interest on identifying the key factors affecting credit spread. One approach is to add macroeconomic variables into the traditional structural model. However, by adding new state variables, the model not only becomes more complicated in the form, but also harder to identify empirical evidence to improve pricing and hedging practice. Another approach is to concentrate on regression models. Because of the simplicity and convenience in incorporating any new state variables, regression is gaining popularity in empirical research for credit spread modeling. Regression models can be divided into two categories: regression on credit spread changes and regression on credit spread levels. Of the first category, there are three major papers: Duffee [1998], Collin- Dufresne, Goldstein and Martin [2001], and Huang and Kong [2003]. Duffee [1998] did the pioneer work on credit spread changes regression. He analyzed the credit spread data indexed by different industry, rating group, and maturity. He used only the interest rate level and slope in the regression, and found that there is a significant negative correlation between short rate change and credit spread change. He achieved an average adjusted R 2 around 17%. Collin-Dufresne et al. [2001] performed similar analysis, but on a lot more variables. They divided corporate bond data by leverage ratio, rating, and maturity, and performed multiple regressions. Among many regression models in the literature, this 101 model appears to be the most complicated. Their basic model included six basic explanatory variables: leverage, interest rate level, interest rate slope, VIX, S&P, jump probability. They achieved around 25% adjusted R 2 . They then performed principal components analysis on the residual and found that over 75% variations are due to the first component. Then they introduced new variables. The total number of variables in final regression is 19, and the adjusted R 2 improved only to 34%. Eventually they acknowledged that they could not identify the factor that contributes to the 75% residual variation, within all the proxy they constructed for liquidity, etc. They claim that the single factor driving the credit spread variation could be attributed to local demand/supply fluctuation. Interestingly, while they introduce new variables, none of these new variables are bond specific; most of them are macroeconomic variables. Huang and Kong [2003] criticized Collin-Dufresne et al. [2001] for not having chosen the best proxies for state variables. So they performed regression on credit spread changes, with similar explanatory variables, while testing multiple proxies for each variable among eight independent variables, and choosing the best one. Also they choose to work with credit spread index OAS data (which they claim as cleaner credit spread) of rating and maturity group. They achieved adjusted R 2 of more than 40% for 5 out of 9 groups. However, the number of observations for each index is merely 67. There is no theoretical support as to why certain proxies for a state variable should perform better than other proxies. And using index data in a short time period might have alleviated the problem. 102 Table 4.1 gives an itemized comparison of the three papers. Duffee [1998] Collin-Dufresne et al. [2001] Huang and Kong [2003] Category Industry, rating, maturity Leverage, rating, maturity Rating and Maturity, total=9 {All sectors, Industrial, Utility, Financial} {<15%, 15-25%, 25- 35%, 45-55%, 55%} Investment Grade: {AA-AAA, BBB-A} {Aaa, Aa, A, Baa} {AAA, AA, A, BBB, BB, B} {1-10 yr, 10-15 yr, 15+ yr} {2-7, 7-15, 15-30} {long (>12 yr), short (<9 yr)} High Yield:{BB, B, C} Data Type Mean corporate yield vs. corresponding. Treasury yield(self constructed index) Corporate yield vs. corresponding Treasury yield Index OAS? N N Y Data Description No option embedded, > 4 yr maturity. Data Range Monthly, Jan-85 to March-95 Monthly, July-88 to Dec-97 Monthly, Jan-97 to July-02 observations At least 25 observations for each bond 67 observations for each index Adj. R-square Around 17% 19% to 25% by leverage ratio >40% for 5 out of 9 17% to 34% by rating group 67% for B 34% after additional variables 60% for BB Table 4.1 Comparison of three papers on credit spread regression Clearly we can see that all these three papers try to improve the explanatory power by either adding more state variables or cherry-picking different proxies for the same state variables. The regression model is fundamentally the same, and the improvement is marginal. 103 Of the second category of credit spread level regression, one major paper is Campbell and Taksler [2003]. They claim that equity volatility in the regression is almost as good as the credit rating variable. What they used in the regression is the excess return (equity return minus market return) volatility for the last 180 trading days, not the historical volatility, or implied volatility from options market. This paper falls into the first category by modeling credit spread changes on individual bonds. We believe there are several benefits focusing on changes: ? Credit spread changes are more relevant to the modeling of credit spread dynamics, since regression on credit spread levels will have a large intercept portion, which is not very informational, because we know that there is always some credit premium associated with the corporate bond yields; ? Regression in credit spread changes is more useful in developing a hedging framework, since we can estimate the sensitivities of the credit spread changes to interest rate, leverage of the company. These sensitivities can be used to derive hedge ratios. ? Individual bond data contain far more information than the indexed data. All the firm-relevant data could enter the modeling, especially the leverage, which is a very important factor in any structural model. However, realizing the drawbacks from simple linear regression, we adopt a more flexible approach: Random Coefficient Regression model. Although the RCR model is not new in statistics (Hildreth and Houck [1968]), it has rarely been used in financial research. 104 4.3 Introduction to Random Coefficient Model The most frequently used linear model in statistics might be the following: iii Xy ?? += , (4.1) where y i is the observed response of dependent variables; X i is the vector of explanatory variables; ? is the vector of coefficients of the linear model; ? i is the error term, and ),0(~ 2 ? ?? N i . For time series data, like those we frequently encounter in financial econometrics, it can be written as: .,...,2,1 , Ttxy tk k tkt =+= ? ?? where y t is the observed random variable, x kt are known explanatory variables, ? k are unknown constants to be estimated, and ? t are the error terms, independently and identically distributed with mean zero, and finite variance. If exact tests of significance are desired, the error terms, ? t , are typically assumed to be normal. In some applications, the constancy of the coefficients, ? k , in consecutive observations may not hold. For example, a particular ? k represents the response of credit spread change for a bond to interest rate, which depends on the demand/supply ratio and market liquidity premium. If both demand/supply ratio and market liquidity premium are relatively stable, the assumption of constancy for ? k might be a tolerable approximation. However, if demand/supply ratio and market liquidity premium vary, but are not 105 observed, assuming ? k as the mean of a random response rate may be better than assuming the response rate to be constant. Consider the following simple extension of model (4.1): iiii Xy ?? += where ii v+= ?? ; 0][ = i vE , ?=]'[ ii vvE , and ? i is uncorrelated with ? i . As before, y i is still the observed random dependent variable and X i are known values of independent variables. In this extension, ? is the mean response of the dependent variable to the independent variables and (? + v i ) is the actual response rate in the ith observation. Combining terms, we have the model: ii iiiii wX vXXy += ++= ? ?? )( (4.2) where 0][ = i wE , iiiiii XXIwwE ?=?+= ']'[ 2 ? An important difference between random coefficient and simple regression model is that the simple linear model assumes the sample is relatively homogeneous. Therefore, if the estimate for ? is zero, then X will be concluded to have no effect on the dependent variable. However, random coefficients may indicate that the effect the results 106 from cancellation of positive effects on some observations with negative effects on other observations. As a result, the randomness of coefficient provides better explanation power even if the mean of the coefficient (? ) is neutral. Dielman, Nantell, and Wright [1980] emphasized that random coefficient models are very useful in analyzing pooled cross-sectional and time series data. In more complex cases, i ? can be parametrically expressed. For example, i ? can be a linear function of several independent variables. While this specification involves additional assumptions, it is essentially the same as the previous simple extension. This functional form is appealing in some cases, especially when there is theoretical basis for the relationship between i ? and those independent variables, and the relationship are of interests of researchers. There are several ways to estimate the model. Details will be given in the next section. 107 4.4 Random Coefficient Model for Credit Spread Changes Huang and Kong [2003] mention that the low explanatory power of theoretical determinants, documented in Collin-Dufresne et al. [2001], could be due to two reasons. The first reason is that the explanatory variables may not be the best proxies to measure the changes in default risk. The second reason is that the current existing corporate bond pricing model might miss some important systematic risk factors. We have a different opinion as to why the simple linear regression model lacks explanatory power. We think the fundamental cause lies in the underlying assumptions of the regression model. When the regression (4.1) is estimated, there is an assumption that the coefficients are fixed. That is, the marginal effect for one unit change in i X has the same effect (? ) on i y regardless of the characteristics of instance i . Suppose credit spread (CS t )is a complex function of interest rate (r t ), firm leverage (lev t ), firm asset volatility (? t ) and other state variables (X t ), which is compatible with most structural models. Given CS t =CS(lev t , r t , ? t , {X t }), we can derive the first order approximation for the change of credit spread: tttt X X CSCS r r CS lev lev CS CS ? ? ? +? ? ? +? ? ? +? ? ? ?? ? ? (4.3) In a short time period, if all these state variables, lev, r, ?, X, do not change dramatically, a simple linear regression model could be used to estimate the coefficients. However, over a relatively long time period, as in Duffee [1998] and Collin-Dufresne et al. [2001], 108 which uses data spanning a 10-year period, this assumption is no longer valid. Because all these gradients themselves are functions of each state variable, they are going to change as well. Specifically, we have: ),,,( ),,,( ),,,( tttt tttt tttt Xrlevh CS Xrlevg r CS Xrlevf lev CS ? ? ? ? = ? ? = ? ? = ? ? Merton [1974] in his seminal paper has calculated the credit spread for a zero- coupon corporate bond in the following form: ?????? ?????? ???? ? ? /)]ln( 2 1 [),( /)]ln( 2 1 [),( )],([ 1 )],([ln 1 )( 22 2 22 1 2 1 2 2 ddh ddh dh d dhrRH +?= ??= ? ? ? ? ? ? ?+??=?= where H is the credit spread; d is a debt ratio measure, defined as d=Be -r? /V; B is the face value of the debt; V is the asset value of the firm; ? is the volatility for the corporate asset process; ? is the maturity of the zero-coupon bond. Then he calculates the credit spread gradient to most state variables as follows: 109 ;0 )( )(' ],[ 2 1 ],[ * 1 *],[ 1 ;0],[ 1 ;0],[ 1 1 2 > ? ? ? == ? ? =?>? 2): r?r, r??, ???, T??. 9 We followed the convention in Collin-Dufresne et al. [2001] to report these statistics. The reported coefficient values are average of the regression estimates for the coefficient on each variable. The t- statistics are calculated by dividing each reported coefficient by the standard deviation of the N estimates and scaled by sqrt(N). 116 Linear Model Variable beta Standard error t intercept -0.006 0.02 -9.45 ?r -0.058 0.21 -7.37 ?lev 0.977 4.77 5.53 ?? 0.005 0.01 8.48 Adj. r 2 0.077 RCR Model Variable beta t intercept 0.05 0.002 -9.23 ?r 18.47 0.68 -1.95 ?lev 295 10.9 -1.38 ?? 1.32 0.049 -1.42 r?r 0.60 0.022 6.02 lev?r 18.59 0.69 -0.42 ??r 0.09 0.003 1.53 T?r 0.84 0.031 1.42 r?lev 12.44 0.46 -0.45 lev?lev 308 11.4 0.91 ??lev 1.68 0.062 -0.32 T?lev 19.41 0.72 1.44 r?? 0.07 0.002 -3.02 lev?? 1.54 0.057 -0.35 ??? 0.01 0.000 3.58 T?? 0.11 0.004 2.30 N 728 Adj. r 2 0.297 Table 4.2 Comparison of RCR vs. linear model These significant interactive terms mean that the level of state variables has a significant impact on the sensitivity of credit spread to these state variables. For example, a positive coefficient for r?r means that when interest rate increases, the sensitivity of credit spread to interest rate change should decrease (because the sensitivity of credit spread to interest rate is negative). In other words, in a higher interest rate environment, credit spread will be less sensitive to interest rate, given everything else unchanged. For the same reason, a negative r?? coefficient means that in a higher interest environment, credit spread will be less sensitive to volatility, when everything else is kept constant. 117 Also a positive ??? coefficient would stand for high volatility sensitivity in high volatility environment. The next table summarizes the relationship we found between levels of state variables and credit spread sensitivities. Sign of beta r CS ? ? (<0) lev CS ? ? (>0) ?? ?CS (>0) Interest + Not Significant (N/S) - Leverage N/S N/S N/S Volatility N/S N/S + Maturity N/S N/S + Table 4.3 Relationship between state variables and credit spread sensitivities These findings validate our assumption that sensitivity should be dependent on state variables. Also we would like to compare these coefficients to structural models, and to validate whether these findings are consistent with theoretical models. We take the most straightforward structural model for credit spread, the Merton [1974] model, for which we have given the derivatives of the credit spread with respect to state variables in section 4.4. Although it is possible to derive the second order derivative of credit spread to validate the relationship we found are consistent with structural model or not, we prefer to demonstrate this in a static analysis, as Merton did in the paper, which will be more intuitive. The following charts show the results of our static analysis. 118 dCS/dr vs. r 0.00% 0.50% 1.00% 1.50% 2.00% 2.50% 3.00% 3.50% 4.00% 0% 5% 10% 15% CS -0.25% -0.20% -0.15% -0.10% -0.05% 0.00% dCS/dr CS dCS/dr Figure 4.1 Credit Spread vs. Risk-free Rate Figure 4.1 shows the credit spread and credit spread sensitivity to interest rate at different interest rate level. This zero-coupon bond is evaluated with 30% leverage, 30% asset volatility and 5-year maturity, which is pretty representative. We can see that while the Merton model predicts the credit spread will be decreasing while the interest rate increases, the credit spread sensitivity to interest rate is an increasing function. However, since the sensitivity measure itself is negative, being an increasing function actually means reduced sensitivity at higher interest rate level, which is consistent with our findings. 119 dCS/dsig vs. sig 0.00% 2.00% 4.00% 6.00% 8.00% 10.00% 12.00% 0% 20% 40% 60% 80% CS 0.00% 0.10% 0.20% 0.30% 0.40% 0.50% 0.60% 0.70% 0.80% 0.90% dCS/dsig CS dCS/dsig Figure 4.2 Credit Spread vs. Volatility Figure 4.2 depicts the credit spread and its sensitivity to volatility at different volatility level. The bond is evaluated at 5% risk-free rate, with 30% leverage and 5-year maturity. From Figure 4.2, we find that credit spread is an increasing function of volatility, and sensitivity to volatility is an increasing function for the most volatility spectrum, from 5% to 60%, and after that, is pretty flat with a slight trend of decreasing. This result is consistent with our findings. 120 dCS/dsig vs. sig 0.00% 0.20% 0.40% 0.60% 0.80% 1.00% 0% 10% 20% 30% 40% 50% 60% 70% 80% dCS/dsig w/30% lev dCS/dsig w/60% lev Figure 4.3 Sensitivity to Volatility at different Leverage In order to test the robustness of the relationship between credit spread sensitivity to volatility and volatility itself, we choose two different setting for maturity and leverage. Figure 4.3 shows the credit sensitivity to volatility for a zero coupon bond with maturity of 15 year, and leverage of 30%, and 60%, at 5% interest rate level. We can see increased maturity make the curve more flat, compared with Figure 4.2. Also increasing leverage makes the yield more flat as well. Since the vast majority of our bonds have maturity less than 15 years, and leverage below 60%, we think the estimated positive coefficient of volatility on sensitivity to volatility is a valid prediction for the majority of these bonds. 121 dCS/dsig vs. r 0 0.05 0.1 0.15 0.2 0% 2% 4% 6% 8% 10% 12% 14% 16% dCS/dsig Figure 4.4 Sensitivity to Volatility vs. Interest Rate Figure 4.4 shows the credit sensitivity to volatility for a zero coupon bond with maturity of 5 year, and leverage of 30%, volatility of 30%, at different interest rate levels. The Merton [1974] model predicts it to be a decreasing function in interest rate, which means that the higher the interest rate is, the lower the sensitivity to volatility will be. The chart is consistent with our coefficient estimator of r??, which is negative. 122 dCS/dsig vs. maturity 0 0.05 0.1 0.15 0.2 0 2 4 6 8 10 12 14 16 dCS/dsig Figure 4.5 Sensitivity to Volatility vs. Maturity Figure 4.5 shows the credit sensitivity to volatility for a zero coupon bond with leverage of 30%, volatility of 30%, at 5% rate level, with different maturities. It shows how maturity change would affect the credit spread sensitivity to volatility. The sensitivity will increase rapidly with respect to maturity till 8 years, and then decrease slightly after maturity passed 8 years. In our estimation, the coefficient is positive, but the significance is not very strong. We predict it is a mixed result of the rapid increasing and slow decreasing. Also Merton?s model is based on zero coupon bond, so if applied to coupon bond, the maturity might be better replaced with duration measure, which is significantly shorter than the maturity. That would explain why we have the coefficient estimator to be positive, which is more inclined to the shorter end of the maturity. 123 4.6.2 Results by Rating and Maturity In this section, we show results for different rating and maturity groups. We have five rating groups: AAA-AA, A, BBB, BB, B and others (below rating B or not rated), and three maturity groups: LONG (maturity >12 years), MEDIUM (12 years >maturity>8 years), and SHORT (maturity < 4years). The total number of combinations is 15. AA_LONG AA_MEDIUM AA_SHORT beta std_error t beta std_error t beta std_error t intercept -0.01 0.001 -5.13 -0.003 0.003 -1.33 -0.01 0.003 -3.34 ?r -0.53 1.01 -0.52 0.72 0.49 1.48 0.43 0.46 0.94 ?lev 8.41 13.12 0.64 8.20 5.13 1.60 2.88 11.59 0.25 ?? -0.39 0.19 -2.00 -0.12 0.06 -2.01 -0.24 0.16 -1.51 r?r 0.10 0.02 5.85 0.05 0.04 1.20 -0.03 0.05 -0.66 lev?r 0.31 0.59 0.53 -0.01 0.86 -0.01 0.07 0.71 0.09 ??r 0.01 0.00 1.61 -0.02 0.01 -1.85 0.01 0.02 0.75 T?r 0.00 0.02 -0.01 -0.07 0.03 -2.25 -0.07 0.06 -1.28 r?lev -0.32 0.38 -0.84 -0.04 0.48 -0.08 0.93 0.67 1.38 lev?lev 4.42 16.88 0.26 -3.94 14.31 -0.28 -39.72 30.33 -1.31 ??lev -0.06 0.09 -0.68 -0.22 0.16 -1.37 0.17 0.26 0.64 T?lev -0.15 0.51 -0.30 -0.39 0.65 -0.60 -0.07 1.17 -0.06 r?? -0.01 0.002 -2.49 -0.004 0.004 -0.95 0.002 0.01 0.25 lev?? 0.07 0.06 1.08 -0.03 0.17 -0.20 0.34 0.24 1.42 ??? 0.00 0.00 0.98 0.002 0.001 1.97 -0.001 0.002 -0.67 T?? 0.01 0.01 2.03 0.01 0.003 4.61 0.02 0.01 1.34 N 42 10 53 Adj. r 2 0.266 0.270 0.233 Table 4.4 RCR coefficients for AA-AAA group For the rating group of AA-AAA, we found that the model performs better (with higher R 2 ) in long maturity group than short maturity group. This is consistent with previous regression model (Duffee [1998]). Also the average explanatory power for this group is below average, which is also consistent with previous regression results. AAA bonds are counted in this group because of the limited numbers in each AAA maturity group. 124 A_LONG A_MEDIUM A_SHORT beta std_error t beta std_error t beta std_error t intercept -0.01 0.002 -4.24 -0.01 0.003 -3.13 -0.01 0.003 -3.73 ?r -1.66 0.68 -2.43 -1.67 0.77 -2.16 -0.33 0.30 -1.10 ?lev 16.91 10.05 1.68 -2.19 7.53 -0.29 4.91 7.65 0.64 ?? 0.00 0.10 0.00 0.08 0.09 0.85 -0.01 0.05 -0.32 r?r 0.11 0.03 3.91 0.12 0.04 3.38 0.11 0.04 2.96 lev?r 0.01 0.51 0.02 1.14 0.92 1.24 -0.02 0.44 -0.05 ??r 0.01 0.00 1.94 0.02 0.01 2.78 -0.01 0.01 -1.44 T?r 0.02 0.02 1.05 0.01 0.05 0.15 -0.04 0.03 -1.47 r?lev -0.21 0.37 -0.57 -0.33 0.76 -0.44 -0.86 0.57 -1.51 lev?lev -1.52 9.49 -0.16 -12.68 14.33 -0.88 -21.83 11.84 -1.84 ??lev -0.01 0.07 -0.19 -0.10 0.19 -0.54 0.13 0.09 1.45 T?lev -0.60 0.45 -1.35 1.26 0.75 1.68 1.56 0.62 2.51 r?? -0.01 0.00 -3.50 -0.02 0.01 -3.19 -0.005 0.003 -1.69 lev?? -0.05 0.06 -0.79 -0.02 0.12 -0.20 -0.03 0.06 -0.45 ??? 0.001 0.0005 2.39 -0.0001 0.001 -0.09 0.002 0.001 2.13 T?? 0.002 0.003 0.59 0.01 0.01 1.12 0.01 0.01 1.28 N 124 35 145 Adj. r 2 0.332 0.298 0.187 Table 4.5 RCR coefficients for A group For the rating group of A, we also found that the performance of our RCR model deteriorates as the maturity decreases. The average explanatory power for long and medium maturity is above and near average (R 2 of 28%), which is also consistent with previous literature (Duffee [1998]). 125 BBB_LONG BBB_MEDIUM BBB_SHORT beta std_error t beta std_error t beta std_error t intercept -0.0175 0.00341 -5.1259 0.0040 0.00686 0.5792 -0.0150 0.00317 -4.7205 ?r -1.4606 1.30092 -1.1228 0.4031 3.15080 0.1279 0.4132 0.67383 0.6108 ?lev 6.3313 14.42570 0.4389 -25.1286 19.48469 -1.2897 -3.3511 15.10461 -0.2210 ?? 0.1922 0.16763 1.1464 0.1464 0.37627 0.3890 0.0037 0.07100 0.0517 r?r 0.1302 0.03744 3.4778 0.0416 0.09990 0.4168 0.1056 0.09241 1.1383 lev?r -0.4979 0.71171 -0.6996 -1.3209 3.10094 -0.4260 -1.2636 1.04115 -1.2089 ??r 0.0309 0.00940 3.2876 -0.0354 0.02867 -1.2358 -0.0130 0.00842 -1.5402 T?r 0.0260 0.05094 0.5094 0.1095 0.15163 0.7223 0.0049 0.13122 0.0373 r?lev 0.3805 0.71955 0.5288 1.4138 1.85106 0.7638 -1.8690 1.80103 -1.0337 lev?lev 7.5851 13.19033 0.5751 34.6082 17.93499 1.9296 -0.8355 20.67328 -0.0403 ??lev 0.1919 0.14222 1.3493 -0.2079 0.29622 -0.7018 -0.2272 0.33345 -0.6786 T?lev -0.4527 0.70592 -0.6413 0.0419 2.22403 0.0189 3.9359 2.19255 1.7880 r?? -0.0187 0.00617 -3.0315 0.0065 0.01899 0.3438 0.0014 0.00847 0.1649 lev?? 0.0901 0.08254 1.0912 -0.1715 0.30587 -0.5607 -0.0434 0.14752 -0.2927 ??? -0.0004 0.00069 -0.5319 0.0033 0.00455 0.7291 0.0031 0.00164 1.8640 T?? -0.0044 0.00747 -0.5856 -0.0151 0.03758 -0.4024 -0.0032 0.00888 -0.3582 N 72 24 126 Adj. r 2 0.3159 0.1873 0.2032 Table 4.6 RCR coefficients for BBB group For rating group BBB, we found that the performance of our RCR model deteriorates as the maturity decreases. The average explanatory power for long and maturity is above average, which is also consistent with previous literature (Duffee [1998]). The regression results for the BBB medium group is far below average, and almost none of these variables are statistically significant, which we suspect is due to limited data problem (only 24 bonds available.) 126 BB_LONG BB_MEDIUM BB_SHORT beta std_error t beta std_error t beta std_error t intercept -0.06 0.01 -11.45 N/A N/A N/A -0.07 0.01 -5.68 ?r -5.18 1.57 -3.29 N/A N/A N/A -4.61 2.35 -1.96 ?lev 21.89 13.93 1.57 N/A N/A N/A -87.91 36.79 -2.39 ?? -0.28 0.08 -3.75 N/A N/A N/A 0.48 0.35 1.40 r?r 0.37 0.04 8.32 N/A N/A N/A 0.34 0.16 2.17 lev?r -1.29 2.21 -0.59 N/A N/A N/A 1.03 2.03 0.51 ??r 0.03 0.01 2.96 N/A N/A N/A 0.01 0.04 0.16 T?r 0.18 0.15 1.19 N/A N/A N/A 0.15 0.16 0.90 r?lev 0.90 0.70 1.29 N/A N/A N/A -1.99 2.28 -0.87 lev?lev 27.62 13.75 2.01 N/A N/A N/A 66.40 38.54 1.72 ??lev -0.78 0.28 -2.80 N/A N/A N/A -0.10 0.83 -0.12 T?lev -1.72 0.66 -2.62 N/A N/A N/A 7.07 3.14 2.25 r?? -0.01 0.00 -3.69 N/A N/A N/A -0.01 0.02 -0.48 lev?? -0.48 0.14 -3.43 N/A N/A N/A -0.66 0.44 -1.51 ??? 0.002 0.001 1.35 N/A N/A N/A 0.01 0.002 3.79 T?? 0.03 0.01 3.83 N/A N/A N/A -0.01 0.03 -0.44 N 24 1 27 Adj. r 2 0.123 N/A 0.104 Table 4.7 RCR coefficients for BB group For the rating group BB, we also see that model became worse when the maturity decreases. And for BB medium group, there is only one bond, so we cannot draw any reasonable conclusion about variable significance. Again the explanatory power is both low for long and short maturity, which could be contributed to limited bond numbers in both categories. 127 B_LONG B_MEDIUM B_SHORT beta std_error t beta std_error t beta std_error t intercept N/A N/A N/A N/A N/A N/A -0.056 0.060 -0.935 ?r N/A N/A N/A N/A N/A N/A -29.005 15.404 -1.883 ?lev N/A N/A N/A N/A N/A N/A 30.750 309.773 0.099 ?? N/A N/A N/A N/A N/A N/A 0.350 2.174 0.161 r?r N/A N/A N/A N/A N/A N/A 1.422 0.609 2.336 lev?r N/A N/A N/A N/A N/A N/A 12.422 10.146 1.224 ??r N/A N/A N/A N/A N/A N/A 0.075 0.183 0.409 T?r N/A N/A N/A N/A N/A N/A 1.372 1.538 0.892 r?lev N/A N/A N/A N/A N/A N/A 46.626 20.449 2.280 lev?lev N/A N/A N/A N/A N/A N/A -419.950 376.543 -1.115 ??lev N/A N/A N/A N/A N/A N/A -3.616 7.040 -0.514 T?lev N/A N/A N/A N/A N/A N/A 22.297 18.849 1.183 r?? N/A N/A N/A N/A N/A N/A -0.291 0.277 -1.047 lev?? N/A N/A N/A N/A N/A N/A 4.618 5.316 0.869 ??? N/A N/A N/A N/A N/A N/A -0.069 0.069 -0.990 T?? N/A N/A N/A N/A N/A N/A -0.162 0.137 -1.186 N 1 1 8 Adj. r 2 N/A N/A 0.500 Table 4.8 RCR coefficients for B and other group The total number of bonds in B and other group are very limited, so we cannot make judgments about model performance in each maturity group. Overall, our model performs best for the A and BBB groups, as well as for longer maturities. These findings are consistent with Duffee [1998], as well as with theoretical structural models for credit spreads. 128 The following table shows the significance level of previously identified interactive terms in each rating and maturity group. Interactive Terms r?r r?? ??? T?? AA_LONG + - Not Significant (N/S) + AA_MEDIUM + - N/S + AA_SHORT N/S N/S N/S + A_LONG + - + N/S A_MEDIUM + - N/S N/S A_SHORT + - + + BBB_LONG + - N/S N/S BBB_MEDIUM N/S N/S N/S N/S BBB_SHORT + N/S + N/S BB_LONG + - + + BB_SHORT + N/S + N/S B_SHORT + - N/S N/S Table 4.9 Summary of RCR coefficients Not surprisingly, we found that for the interactive terms, which were constructed in the RCR model, the significance levels and signs of the coefficient estimators are very consistent for each rating and maturity group. If we remove the three groups with one sample ecah, the r?r term is significant for 10 of 12 groups, which means that the interest rate level has a positive impact on the credit spread sensitivity on interest rate, no matter what rating group or maturity category. Also the impact of interest rate level on credit spread sensitivity on volatility is very consistent. 129 4.6.3 Non-Constancy of Credit Spread Sensitivities The non-constancy of credit spread sensitivities would naturally be embedded in their dependence on state variables in the RCR model. However, we would like to see how they change over time, and compare it to the simple linear regression sensitivity estimators, and find out why the RCR estimators provide better accuracy. Let?s take one bond as example, the bond with CUSIP of "001765AE", one of American Airlines' long term bonds, and depict its random coefficients and constant coefficients. The following three figures show the comparison of regression coefficients with respect to interest rate changes, leverage changes, and volatility changes. sensitivity to r -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 1 990 06 1 990 10 1 991 02 1 991 06 1 991 10 1 992 02 1 992 06 1 992 10 1 993 02 1 993 06 1 993 10 1 994 02 1 994 06 1 994 10 1 995 02 1 995 06 1 995 10 1 996 02 1 996 06 1 996 10 1 997 02 1 997 06 1 997 10 beta_dr_linear beta_dr_RCR Figure 4.6 Coefficient for ?r in RCR vs. Linear Model 130 sensitivity to lev 0 1 2 3 4 5 6 7 8 9 199006199009199012199103199106199109199112199203199206199209199212199303199306199309199312199403199406199409199412199503199506199509199512199603199606199609199612199703199706199709199711 beta_dlev_linear beta_dlev_RCR Figure 4.7 Coefficient for ?vol in RCR vs. Linear Model sensitivity to sig -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 1 990 06 1 990 10 1 991 02 1 991 06 1 991 10 1 992 02 1 992 06 1 992 10 1 993 02 1 993 06 1 993 10 1 994 02 1 994 06 1 994 10 1 995 02 1 995 06 1 995 10 1 996 02 1 996 06 1 996 10 1 997 02 1 997 06 1 997 10 beta_dvix_linear beta_dvix_RCR Figure 4.8 Coefficient for ?lev in RCR vs. Linear Model There are three major findings from the graphs: 1. Sensitivity to ?r does change over time. In Merton?s model, increased interest rate would increase the risk neutral drift term, thus decrease the default probability, and shrink the credit spread. In reality, Fed generally lowers interest rate to stimulate economy when there is a recession, which is the case during 1990-1992. Generally higher credit spreads are observed during a recession. That is the main reason for negative correlation between interest rate and credit spread. However, 131 what about during times of economic recovery or boom? It would be interesting to compare Figure 4.6 which depicts the sensitivity of credit spread to interest rate, to Figure 4.9, the history of 3-month Treasury rate, a close reflection of Fed?s policy on funding rate. When the economy is recovering, lowering interest rate would have less effect on credit spread. That is exactly the case we found during 1992-1993, when the Fed continued lowering the short interest rate, and the sensitivity of credit spread to the interest rate is close to zero. Also when economy is booming, the Fed is likely to raise the interest rate, and that seems to have little effect on the credit spread. That is likely the case for 1994-1995. 2. Sensitivity to ?? also changes over time. In structural model, increase in volatility would increase the default probability, and thus widen the credit spread. However, comparing Figure 4.8 with 4.10: the history of VIX volatility index, we found that while volatility is high both in the early 90?s and the late 90?s, their impact on credit spread sensitivity are quiet different. One explanation for this could be that during a recession, volatility is a bad thing, because it is likely that the volatility is a result of dropping equity, and investors will be really concerned with a volatility spike. However, when the economy is booming, it is likely that high volatility is introduced by rising stock prices, and investors are less likely to require high credit spread for this ?good volatility?. 132 3m Treasury Rate 0 1 2 3 4 5 6 7 8 9 Jan-90 J ul -90 J an - 91 J ul -91 J an - 92 Jul - 92 Jan-93 Jul - 93 J an - 94 J ul -94 J an - 95 Jul - 95 Jan-96 Jul - 96 J an - 97 J ul -97 3m Figure 4.9 Three-Month Treasury Rate from 1990 to 1997 vix 0 5 10 15 20 25 30 35 40 19 9 005 19 9 010 19 9 103 19 9 108 19 9 201 19 9 206 19 9 211 19 9 304 19 9 309 19 9 402 19 9 407 19 9 412 19 9 505 19 9 510 19 9 603 19 9 608 19 9 701 19 9 706 19 9 711 Figure 4.10 VIX index from 1990 to 1997 Of the three volatility measures we used, our results show that VIX is better than both history volatility and excess volatility, which is unanticipated. Originally we thought that since VIX is a broad market volatility index, replacing it with company specific volatility should improve our results. The reason for this phenomenon might be that credit spread response is more sensitive to market perception of risk than to historical 133 volatility. Also we tried the excess return volatility, which Campbell and Taksler (2003) claims to have significant explanatory power in regression of credit spread levels. The results are disappointing, and the adjusted R 2 is comparable to historical volatility, but not as good as VIX index. The reason might be that the credit spread itself already has a build-in premium associated with the standard deviation of excess return, but the change of credit spread is not sensitive to its change, so the regression on credit spread levels and changes will have different explanation. 134 4.7 Conclusions and future work By using the RCR approach, we not only model the dynamics of credit spread sensitivities in a more consistent way with current structural model, but also achieve more explanatory power than simple linear model. Our contributions are the following: 1. The first paper to use RCR model on credit spread data; 2. The first paper to explicitly model the credit spread sensitivities with dependence against state variables, and empirically validate the dependence relationship predicted by Merton?s model; 3. Higher explanatory power is achieved without adding new independent state variables. In this case, we increased the adjusted R 2 from 8% to 30%. Obviously, there are still some unanswered questions remaining in our work. We would like to pursue future research in the following directions: 1. We can see from our results analysis from section 4.5, the theoretical sensitivity changes are not always linear with respect to state variables (Figure 4.5), and when there is a strong no linear relationship, our predictions of coefficient are generally weaker. So can we change the functional form in the regression model for sensitivity parameters and achieve better explanatory power? 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